Downhole method for determining geologic permeability

ABSTRACT

A method for predicting formation permeability by measuring diffusional tortuosity in several directions by pulse gradient NMR. The method comprises evaluating an anisotropic diffusion coefficient by pulsed gradient NMR, determining diffusional tortuosity from the restricted diffusion data, supplementing the NMR results with resistivity and sonic logging data, measuring anisotropic tortuosity and porosity by resistivity and sonic data and combining all components in a single fitting model. The 11-coefficient model is trained to recognize the true values of permeability by comparing the real oil permeabilities measured in a library of oil-carrying rock cores with the NMR, resistivity and sonic correlates. The fitting coefficients are extracted by minimizing the discrepancy between the laboratory measured permeabilities and the predicted values combining all rapid logging information components with the agreement-maximizing weights.

BACKGROUND OF THE INVENTION Technical Field

The present disclosure relates to a method, system and apparatus forevaluating permeability anisotropy in hydrocarbon-carrying geologicalformations.

Description of the Related Art

The “background” description provided herein is for the purpose ofgenerally presenting the context of the disclosure. Work of thepresently named inventors, to the extent it is described in thisbackground section, as well as aspects of the description which may nototherwise qualify as prior art at the time of filing, are neitherexpressly or impliedly admitted as prior art against the presentdisclosure.

Permeability is an important parameter in evaluating reservoir qualityand fluid flow of a hydrocarbon-containing subterranean geologicalformation or samples taken therefrom. Evaluating permeability anisotropyin layered or fractured reservoirs is important for production andstimulation planning. The term “permeability anisotropy” describes theratio between the vertical and horizontal permeability of the geologicalformation, the latter being frequently larger (See J. R. Lishman, “CorePermeability Anisotropy,” J. Can. Pet. Technol., vol. 9, no. 02, p. 7,April 1970). Permeability anisotropy evaluation is frequently conductedusing either 3D images of rock samples, e.g., x-ray images (H. Sun, S.Vega, and G. Tao, “Analysis of heterogeneity and permeability anisotropyin carbonate rock samples using digital rock physics,” J. Pet. Sci.Eng., vol. 156, pp. 419-429, July 2017) or by extracting two differentcores (R. Meyer, “Anisotropy of sandstone permeability,” CREWES ResearchRep., vol. 14, pp. 1-12, 2002) one parallel and one perpendicular to thelayering direction (e.g., lamination of shale embedded in coarse sands).The development of a faster technique that can be implemented in thefield can be of great significance in many science and engineeringapplications.

One such method is modeling permeability anisotropy by measuringdiffusion coefficients using nuclear magnetic resonance (NMR). Such atechnique may be applicable at both laboratory scale or by using an NMRdownhole tool in the field. The restricted fluid self-diffusion measuredin rocks by NMR equipment can be utilized to estimate the diffusiontortuosity, which in turn is strongly related to permeability. Since amagnetic gradient can be applied in different directions, the anisotropyof diffusion tortuosity and thus permeability can be evaluated.

Vertical boreholes are fortified by steel casing, and in order to extendhydrocarbon production operations the casing is perforated, andhorizontal or arbitrary angle channels are drilled in the rock. Thesedaughter channels are not cased, but instead form the starting sites forfracturing or acid treatment. The identification of maximum and minimumpermeability from NMR diffusion measurements aligns the main verticalborehole casing perforations with the maximum permeability zones,improving oil and gas productivity.

Real-time NMR logging applies to the measurement of permeabilityanisotropy during drilling both vertical and horizontal wells.Evaluating permeability anisotropy in hydrocarbon-carrying formations isimportant in applications such as for fluid flow simulation andproduction planning in hydrocarbon reservoirs and aquifers. The latteris particularly important where petrophysical characteristics vary withdepth. Current techniques for evaluating permeability anisotropy requireeither 3D imaging of rock cores or extraction of cores from twodifferent directions; one parallel and one perpendicular to thelayering. Both techniques are time-consuming and limited to laboratoryinvestigation. Consequently, timely evaluation of in-situ permeabilityanisotropy in the wellbore remains a challenge.

The importance of anisotropy analysis follows the engineeringfundamentals of both (a) hydrocarbon flow under stimulating conditionsand (b) hydraulic fractioning stimulating hydrocarbon extraction.Regarding the aspect (a), the hydrocarbon strata exist under lithostaticpressure that increases nearly linearly with the depth of the formationand respective hydrocarbons. Hydrocarbon-rich strata follow the Laplacelaw of isotropic pressure distribution. Thus, an uncompensatedhorizontal pressure component directed toward the borehole develops in acompressed formation.

Viscous fluids escape towards the downhole following the Darcy-Weisbachequation:

$\begin{matrix}{{\frac{\Delta p}{L} = {\frac{128}{\pi} \cdot \frac{\mu Q}{D_{c}^{4}}}},} & (1)\end{matrix}$

where μ is the dynamic viscosity of the fluid (Pa·s=N·s/m²=kg/(m·s)); Qis the volumetric flow rate of the hydrocarbon, used to measure flowinstead of mean velocity according to Q=π/4Dc²(v) (m³/s), ΔP is thehorizontal component of the lithostatic pressure (N/m²), L is the lengthof the escape route (m). The equation (1) is entirely applicable tonatural gas and to richer oil deposits capable of forming continuouspaths that pass the pressure from site to site. Equation (1) is lessapplicable to depleted deposits in need of stimulation, since the lackof hydraulic continuity violates the assumptions of Darcy's derivation.

The viscous flow of hydrocarbons out of the formation is a function ofthe inverse 4^(th) power of the equivalent pore diameter at a givenhorizontal pressure difference and therefore increasing the equivalentpore diameter increases the formation's productivity. The length Ldecreases productivity. Considering a rock unit volume and movement of ahydrocarbon element across this unit volume, it is apparent thatproductivity is sensitive to:

-   -   1) overall porosity (the volume, number and distribution of        hollow pores in the rock),    -   2) tortuosity τ (the ratio of the actual path covered by a        differential element of fluid in the rock volume to the        straight-line length of the volume),    -   3) formation resistivity factor F (the ratio of electric        resistance in a fluid-saturated porous media and the bulk        permeating fluid), and    -   4) restrictivity (the number and extent of “bottleneck”        situations, with cul-de-sacs considered to be the extreme        chokepoints).

Tortuosity is known to be inversely proportional to overall porosity,and restrictive elements in the structure are specific to the formation(See Rezaee M R, Motiei H, Kazemzadeh E. A new method to acquire mexponent and tortuosity factor for microscopically heterogeneouscarbonates. Journal of Petroleum Science and Engineering. 2007 Apr. 1;56(4):241-51 incorporated herein by reference in its entirety).

Returning to the analysis by the Darcy equation, it is apparent that thecalculated productivity may vary by orders of magnitude as a function ofdiverse aspects of formation microstructure. For example, a sample withthe pore diameter 0.5 vs. 1 in a control, tortuosity 2 vs. 1 in thecontrol and 50% of all paths ending in chokepoints vs 0% in the controlis at least 64-fold less productive than the control. At lowerporosities, the effects can be more dramatic. These observationsnecessitate measuring the petrophysical parameters before expandingdrilling operations, at the earliest stage of exploration (e.g., beforesteel encasing and branching attempts) and preferably by a rapid in-situlogging method.

In the EOR (Enhanced Oil Recovery) contexts, the Darcy physics areapplicable to the steam, flooding brine, flooding acid and fracturingfluids. The pressure loss during these procedures and the expected yieldof the detached, liberated or solubilized hydrocarbon moving withheterogenous or aqueous continuous phase can be computed with theknowledge of permeability as well.

Typically, stimulated extraction of hydrocarbons involves hydraulicfracturing. Fracturing occurs when pressurized fluid (e.g., saline),mixed with proppants and viscosity modifying gels (to prevent proppantsedimentation) contacts pores and micro-defects on carbonate orsandstone surfaces facing the cased drilling pipe. Assuming no suchinitial micro-defects, it is difficult to propose an alternativemechanism of macroscopic fracture formation. The pores are typicallywithin the range of compressive elastic deformation, and many are closedby lithostatic pressure. The acting forces are approximately balanced inboth vertical and horizontal direction around a closed pore in acompressed rock. In the proximity of the drilling borehole, thehorizontal component of pressure acting on a pore is asymmetric. Ahorizontal force acting on the pore's “bottom” on the rock's side is notcompensated by an equal force on the fracturing “mouth” pore side. Thereaction forces developing in this context act against the compressiveeffect of the vertical pressure and combine with the forces ofhydrostatic pressure gradient, capillary pressure and the additionalpore-opening pressure gradient component provided by fracturing pumps.Together, these parameters define a fracture gradient (FG), and itscorrect prediction is important for economically viable stimulatedhydrocarbon production (See Eaton B A. Fracture gradient prediction andits application in oilfield operations. Journal of petroleum technology.1969 Oct. 1; 21(10):1-353; Liang, Q. J., 2002, January. Application ofquantitative risk analysis to pore pressure and fracture gradientprediction. In SPE Annual Technical Conference and Exhibition. Societyof Petroleum Engineers; Zhang J, Yin S X. Fracture gradient prediction:an overview and an improved method. Petroleum Science. 2017 Nov. 1;14(4):720-30—incorporated herein by reference in its entirety). Thesubject is of significant impact due to a possibility of over-fracturingand collapse of the tunnel's walls surrounding a tilted or a horizontalpipe, sourcing the fracturing fluids. Typically, in the overall schemeof stimulated production such horizontal paths branch off the mainvertical shaft and the fracturing takes place in these branches(“perforation scheme” means perforating the casing of the centralvertical downhole to develop the side routes). The fracturing process,however, is self-propagating, once it starts at a given FG. The primaryfissures engender secondary fissures, and flood-work stability wouldrequire maintaining the weakened tunnel under a higher working fluidpressure all the time (See Asgian, M. I., Cundall, P. A. and Brady, B.H. G., 1995. The mechanical stability of propped hydraulic fractures: anumerical study. Journal of Petroleum Technology, 47(03), pp. 203-208;Bernt S. Aadnoy, Reza Looyeh, in Petroleum Rock Mechanics (SecondEdition), 2019—each incorporated herein by reference in its entirety).This leads to increased fluid loss, aquifer contamination and greaterecological concerns. Thus, it is important to correctly estimate allcomponents of the fracturing gradient, including rock wettability, poremouth sizes and geometry to correctly plan the borehole family, thedepth, angles and number of branches, the working pressure of fluids.These objectives superimpose on the previously considered permeabilitymanagement. Therefore, determining the porosity characteristics offormation rocks as soon as drilling allows is a pivotal step in runninga stimulated drilling process with high ROI (return on investment).

Proton Nuclear Magnetic Resonance is applicable for the measurement ofporosity and tortuosity in-situ. The initial formula for extraction ofdiffusion coefficients from the NMR pulse sequence data are provided inStejskal, E. O.; Tanner, J. E. in the publication titled “Spin DiffusionMeasurements: Spin Echoes in the Presence of a Time-Dependent FieldGradient” in the Journal of Chemical Physics, 1965, 42 (1): 288incorporated herein by reference.

$\begin{matrix}{\frac{S({TE})}{S_{0}} = {\exp\left\lbrack {{- \gamma^{2}}G^{2}{\delta^{2}\left( {\Delta - \frac{\delta}{3}} \right)}D} \right\rbrack}} & (2)\end{matrix}$

where

-   S₀ is the signal intensity without the diffusion weighting,-   S (TE) is the signal with the magnetic field gradient,-   γ is the gyromagnetic ratio,-   G is the strength of the gradient pulse, δ is the duration of the    pulse,-   Δ is the time between the two pulses, and finally,-   D is the diffusion coefficient.

The pulsed NMR of Stejskal et. al. can be fine-tuned for anisotropicmaterials and the 13-interval APGSTE sequence is a preferred methodbecause it takes into account magnetic field pulses in multipledimensions, therefore measuring diffusion coefficients in differentdirections. The signal attenuation (S/S₀) for the 13-interval APGSTEsequence is given by a variation of the Stejskal and Tanner equation(2):

$\begin{matrix}{\frac{S}{S_{o}} = {\exp\left\{ {- {\quad{D_{R}\gamma^{2}\left. \quad\left\lbrack {{{\delta^{2}\left( {{4\Delta} + {6\lambda} - \frac{2\delta}{3}} \right)}g_{a}^{2}} + {2{{\lambda\delta}\left( {\delta_{1} - \delta_{2}} \right)}g_{a}g_{0}} + {\frac{4}{3}\lambda^{3}g_{0}^{2}}} \right\rbrack \right\}}}} \right.}} & (3)\end{matrix}$

where

-   S₀=NMR signal in the absence of applied magnetic field gradients    (μV).-   D_(R)=Restricted self-diffusion coefficient of the fluid (m²/s),-   Δ is the duration between two applied magnetic field ‘Diffusion    Time’ (msec),-   δ is the duration of the applied magnetic field gradient (msec).-   γ is the gyromagnetic ratio of the nucleus being studied (=2.68×10⁸    Hz/T for ¹H nucleus).-   g_(a) is the strength of the applied magnetic field gradient (T/m),-   g_(o) is the strength of the internal magnetic field gradient (T/m).-   λ is the time between the first two (π/2) RF pulses (msec), and-   and δ₁ and δ₂=pre- and post-pulse time respectively (msec).

The publication by Dvinskikh S V, Furo I. (See “Measurement of theprincipal values of the anisotropic diffusion tensor in an unorientedsample by exploiting the chemical shift anisotropy: 19F PGSE NMR withhomonuclear decoupling”. Journal of Magnetic Resonance. 2001 Jan. 1;148(1):73-7—incorporated herein by reference in its entirety) teachesPGSE with dipolar decoupling extended to polycrystalline solids andunoriented liquid crystals. Decoupling suppresses dipolar dephasing notonly during the gradient pulses but also under signal acquisition sothat the detected spectral shape is dominated by the chemical shifttensor of the selected nucleus. The decay of the spectral intensity atdifferent positions in the powder spectrum provides the diffusioncoefficient in sample regions with their crystal axes orienteddifferently concerning the direction of the field gradient. Hence, onecan obtain the principal values of the diffusion tensor. The method isdemonstrated by 19F PGSE NMR with homonuclear decoupling in a lyotropiclamellar liquid crystal. The method is not directed to hydrocarbonreservoirs and is not 1H NMR.

US2018348396 discloses a method for determining permeability insubsurface anisotropic formations. An effective permeability K (or apermeability component) can be determined using conventional logdata-derived permeabilities (e.g., using NMR, RDT, sonic, and/orresistivity logs). Permeability components can be determined usingresistivity anisotropy data obtained from MCI measurement data and theeffective permeability K from conventional permeability logs. Thisdisclosure focuses on fluid filling pore-space behavior. Also, all theNMR techniques used in this study are either relaxation experiment (T1and T2) or multi-dimensional techniques (COSY, EXSY, spin-diffusion).The publication does not teach rock anisotropy but rather fluidproperties.

US2017285215 discloses pore fluid phase behavior measurements. Laplaceinversion algorithms may be used to process adsorption and NMR data toobtain a multi-dimensional distribution of adsorption and NMR relaxationparameters, which contain the key information regarding themicro-porosity and gas storage mechanism for even a single-componentfluid. The reference does not focus on measuring of tortuosity and useof APGSTE 13 pulsed NMR sequence.

WO19232299 discloses a method for assessing pore-throat sizedistribution and permeability of porous media. The method includesusing:

-   -   (a) NMR data which correlate to hydraulic tortuosity (rH),    -   (b) an electrical constriction factor (CE) from the measurements        that correlates to hydraulic constriction factor (CH), and    -   (c) pore-size distribution (rP) and effective pore radii        calculated from the NMR data, and    -   (d) a connected porosity (cpc) estimated from the NMR data, in        porous media where total pore volume is approximately equal to        the connected pore volume or estimated from three-dimensional        pore-scale images of the section of the porous media.

The NMR method comprises measuring the relaxation time of the transversemagnetization decay (T2) using a 2 MHz Magritek NMR Rock Core Analyzerwith a CPMG pulse sequence. The CPMG sequence is distinct from APGSTE13. The image analysis is relatively cumbersome. The disclosureestimates bulk rock properties based on pore size distribution (noconsideration of permeability anisotropy). The patent teaches the use ofT2 measurements by applying PFG family NMR signal sequence measured atdifferent directions.

RU2292541C1 discloses applying a visible mark on the specimen to measurean angle of rotation of the specimen around its axis, saturating thespecimen with a proton-containing liquid, setting the specimen into thespectrometer, making the visible mark coincident with the directionalong the chosen axis of the laboratory coordinate system, setting thespecimen in rotation around its axis with a uniform speed until therotation angle between the mark and direction along the laboratorycoordinate axis reaches 360°. For every current rotation angle, themeasurement of the current angular amplitude of the spin-echo signalfrom the proton-containing liquid is performed before and after theevent. The anisotropy coefficient and the characteristics of the mainaxes of anisotropy of rock pore follows the data obtained for a givenvalue of the pulse gradient. The method of RU2292541C1 is based onrotating sample (not changing gradient direction) around its axis with afixed speed. The patent does not mention determining diffusiontortuosity which requires the measurements of restricted diffusioncoefficient at different diffusion times (Δ).

US2004090230, incorporated herein by reference in its entirety,discloses an apparatus and method for determining porosity andpermeability, as well as the characterization of fluids in theformation. The disclosure teaches means for determining anisotropicpermeability where vertical permeability (K_(z)) differs from horizontalpermeability (K_(xy)). The invention relies on Pulsed Field Gradient(PFG) pulse sequence. In a two-dimensional context, two differentimaging techniques are commonly utilized, Fourier Imaging (FI) andProjection Reconstruction (PR), sometimes known as the back-projectionimaging or inverse radon technique. In PR technique, the object to bestudied is first subjected to a magnetic field. A one-dimensional fieldgradient is applied at several angles, and the NMR spectrum is recordedfor each gradient angle. A first spectrum is recorded with the gradientat zero degrees to the +Y axis. A second spectrum is recorded with thegradient now at a one-degree angle to the +Y axis. The process isrepeated for the 360 degrees between 0 degrees and 359 degrees. Therecorded data can be converted into an image by a 2D Fouriertransformation in cylindrical coordinates. The publication teachesestimating the permeability based on imaging. The method requires atleast two MRI images, and fluid should be flowing while measuring. Thereference does not discuss permeability anisotropy and diffusiontortuosity. The technique has a limited resolution that preventscapturing small features (<100 um).

US2007238969 discloses diffusion-based magnetic resonance methods forcharacterizing bone structure. The method relies on time-dependentdiffusion coefficients along two different directions measured with acompensated pulsed field gradient sequence. Fractional anisotropy isestimated by a surface-to-volume ratio tensor calculated from PFGtime-dependent diffusion experiments. This method includes estimatingfractional anisotropy of bone structure by surface to volume (S/V)—shorttime regime—of PFG NMR diffusion experiment, not APGSTE 13. The use ofdiffusion tortuosity is not considered and applies not to hydrocarboncarrying formations.

US2019204248 discloses a method for nuclear magnetic resonance diffusionmeasurements, differentiating between intrinsic bulk diffusivity of thesubstance and the reduced diffusivity of the substance caused byrestricted diffusion. The d-PFG pulse sequences are applied multipletimes while the area parameters (q1) and (q2) are held constant and agradient angle between the pairs of gradient pulses is varied. The firstpair 304 is applied along a single direction (e.g., x-axis) and thesecond pair 306 is applied along a different direction (e.g., y axis).As the d-PFG pulse sequences are applied, the second direction is variedand the gradient angle between the pairs thus also varies. A plot of theNMR signal for different values of the gradient angle can potentiallyshow modulation due to time dependent diffusion and diffusionanisotropy. This disclosure extract pore size distribution from thestructure. This disclosure is using d-PFG to quantify the diffusionanisotropy. The d-PFG pulse sequence is applied multiple times to afluid within a heterogeneous porous medium, using a variable mixing time(Tm) between the two diffusion periods (Δ1) and (Δ2). The publicationteaches no specific application for permeability anisotropy.

It is therefore another object of the present disclosure to provide animproved method of tortuosity and porosity assessment in-situ underdynamic drilling situations in the hydrocarbon-bearing formationsundergoing stimulation by contact with the high-pressure fluids.Real-time responses at a larger scale are linked to hydrocarbon outputvia relationships of economic efficiency. Another objective islarge-scale modeling of the entire basin based on permeability atseveral locations.

SUMMARY OF THE INVENTION

According to a first aspect, the invention relates to an NMR-basedmethod of restricted diffusion coefficient measurement under actualdownhole drilling conditions, wherein the diffusion of protons in afluid such as a drilling mud is coupled to porosity and tortuosity.

According to a second aspect, an NMR-based method utilizes gradientfields producing well-defined voxels of the environment's volume arounda drilling installation comprising a downhole NMR apparatus. A Fouriertransforms the spin echoes in each voxel to the coordinate, diffusioncoefficient and strength of proton signals, indicating the spatialdistribution of tortuosity and porosity in the formation.

According to a third aspect, an NMR-based method utilizes a sequence of90-degree, 180-degree or partial degree radiofrequency pulses, magneticgradient pulses of any polarity and direction and calculations capableof interpreting the spin-echo data and converting it in the diffusioncoefficients.

According to a fourth aspect, the NMR-derived diffusion coefficientsmeasured in the mud fluid bulk and in the adjacent 0-5 mm “skin” of theporous formation are compared, producing the restricted diffusioncoefficients.

According to a fifth aspect, a critical length of time is allocated forobserving asymptotically constant values of the restricted diffusioncoefficients, and only these asymptotic values are used to computetortuosity.

According to a sixth aspect, the bulk and restricted diffusioncoefficients are computed in different directions and are related,forming the anisotropic tortuosity values.

According to a seventh aspect, the information of NMR analysis aloneapplies to predicting the formation permeability.

According to an eighth aspect, the resistivity and acoustic loggingsupplements the NMR-only information, also used to predict anisotropictortuosity and porosity.

According to a ninth aspect, all sources of high-throughput logginginformation are combined in a single regression model, relating theobserved permeabilities in the hydrocarbon-carrying cores and thecorresponding NMR, resistivity and acoustic correlates.

According to a tenth aspect, the regression model is trained andimproved by maximizing the agreement between the laboratory values ofpermeability and the predicted values based on combining NMR,resistivity and sonic signals with the fitted coefficients. Thecondition of minimal disagreement between the model and laboratoryexperiments defines values of the fitted coefficients.

According to an eleventh aspect, the samples for the study in the tenthaspect are procured globally and reflect diverse hydrocarbon reservoirs.

According to a twelfth aspect, 50% of data points are retained as atesting set allowing to extrapolate the training predictions and adjustthe mathematical form to achieve good generalization for a new set ofconditions (a new reservoir not a part of prior experimentation andmeasuring).

According to a thirteenth aspect, economic models are provided thatrelate the structure of the reservoir, permeabilities established by theinventive method, location of the drilling installation and the costs ofthe process.

According to a fourteenth aspect, the economic models include theperforation scheme of the vertical casing, the number of branchingchannels, the angles and the lengths of the branching channels, theextent of fracturing—all estimated with the metrics solely based ondiffusional tortuosity or comprising diffusional tortuosity as anessential component of a more comprehensive model.

According to a fifteenth aspect, the planning of a branched well, thestructure of hydrocarbon reservoir, permeability and wettability of thewell's walls are all considered by a software based on the regressionmodel with diffusional tortuosity as a sole or essential feature.

The foregoing paragraphs have been provided by way of generalintroduction and are not intended to limit the scope of the followingclaims. The described embodiments, together with further advantages,will be best understood by reference to the following detaileddescription taken in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete appreciation of the disclosure and many of the attendantadvantages thereof will be readily obtained as the same becomes betterunderstood by reference to the following detailed description whenconsidered in connection with the accompanying drawings, wherein:

FIG. 1: The basic components of a gradient NMR installation.

FIG. 2: The scheme of 90 degree flipping and synchronization for theglobal magnetization vector.

FIG. 3: The scheme of 180 degree flipping and synchronization for theglobal magnetization vector.

FIG. 4A. The initial orientation of the magnetization vector up-field.

FIG. 4B. The initial focusing after the completion of 90 degree “flip”.

FIG. 4C. Relaxation of spins post 90-degrees focusing. Faster elementslose coherence fist, while the lagging remain focused.

FIG. 4D. After RF irradiation from the opposite direction the formerinvisible elements (in 4C) become leading, face the detector coil andproduce a signal. The former visible elements become lagging and areinvisible.

FIG. 4E. From the initial 90 degree “flip” of FIGS. 4B and 4C, theglobal vector first turns by 90 degrees vs, 4B and 4C (or 180 degrees vs4A) and next additional 90 degrees, arriving at the arrangement asshown. After disconnection of RF energy, the individual magnetizationvector continues synchronizing in the global transverse magnetizationfield M (as they do in B₀ in 4A), reaching a refocusing point later intime.

FIG. 4F. From the initial 90 degree “flip” of FIGS. 4B and 4C, theglobal vector first turns by 90 degrees vs, 4B and 4C (or 180 degrees vs4A) and next additional 90 degrees, arriving at the arrangement asshown. The lagging elements continue to move in the transverse magneticfield M even when the RF energy is shut down, in the same sense as theyconverge in B₀ field in 4A. The field M acts more on the laggingelements and less on the leading elements, contributing to focusing ofthe signal in phase, observing “echo”, even when RF is turned off.

FIG. 5. Dependence of relaxation times T1 and T2 on the chemical nature,free vs. bound status and molecular weight of the molecules in themagnetized environment.

FIG. 6A: General scheme of PGSE (pulse field gradient echo) formeasuring molecular diffusion.

FIG. 6B: The relation of RF and magnetic gradient pulses in thediffusional measurements.

FIG. 6C: Development of decoherence and the subsequent re-focusing inthe nuclei assembly as a function of RF pulses (hollow rectangularvertical bars) and magnetic gradient pulses (blue rectangular horizontalbars).

FIG. 6D: Accumulation of data points in the diffusion measurements byNMR.

FIG. 7: Diffusion ordering spectroscopy NMR (general scheme).

FIG. 8A: Transformation of the f1-G data into DOSY data. The primarydata: the axis S corresponds to the chemical shifts, the axis Mcorresponds to the intensity of magnetic pulse gradient. Differentspecies show different rate of extinction.

FIG. 8B: Transformation of the f1-G data into DOSY data. The data 8A arereorganized. The axis D corresponds to the chemical shifts, the axis f1corresponds to the intensity of magnetic pulse gradient.

FIG. 9: A scheme of stimulated echo sequence (STE) NMR experiment.

FIG. 10: A scheme of longitudinal encode-decode or “longitudinal eddycurrent delay” echo sequence NMR experiment.

FIG. 11: Bipolar gradient longitudinal encode-decode BPP-LED pulsesequence NMR experiment.

FIG. 12: Alternating Pulsed Field Gradient Stimulated Echo NuclearMagnetic Resonance (APGSTE) sequence.

FIG. 13: Representation of the Application of NMR Pulse Field Gradientin Different Directions (modified afterhttp://mri-q.com/gradient-coils.html).

FIG. 14: Restricted diffusion coefficient of water as a function ofobservation time for Indiana (2). The dashed line corresponds to theasymptotic regime.

FIG. 15: Restricted diffusion coefficient of water as a function ofobservation time for Indiana (1).

FIG. 16: Diffusion tortuosity versus permeability for the four samples.The best fine curve is shown in dashed blue. Sample A and B refer totight and permeable samples whose 3D pore network is shown in FIGS. 17Aand 17B.

FIG. 17A: 3D image of a pore network extracted from micro CT scanningfor tight Indiana limestone.

FIG. 17B: 3D image of a pore network extracted from micro CT scanningfor permeable Indiana limestone.

DETAILED DESCRIPTION OF THE EMBODIMENTS

Embodiments of the present disclosure will now be described more fullyhereinafter with reference to the accompanying drawings, in which some,but not all of the embodiments of the disclosure are shown.

The present disclosure will be better understood with reference to thefollowing definitions.

As used herein, the words “a” and “an” and the like carry the meaning of“one or more”. Additionally, within the description of this disclosure,where a numerical limit or range is stated, the endpoints are includedunless stated otherwise. Also, all values and subranges within anumerical limit or range are specifically included as if explicitlywritten out.

As used herein, the term “optional” or “optionally” means that thesubsequently described event(s) can or cannot occur or the subsequentlydescribed component(s) may or may not be present (e.g. 0 wt %).

As used herein, the term “flip” and “flip angle” refers to the change inthe orientation of the nuclear magnetic momenta in the primary magneticfield due to the absorption of radiofrequency pulse. The resultingmacroscopic magnetic momentum gyrates with the resonant Larmorprecession rate producing the signal detectable by a separate or thesame radiofrequency coil (that produced the said pulse).

As used herein, the terms “collective”, “global’ and “macroscopic” areequivalent when referred to a transverse magnetization vector M, alsomentioned as “net magnetization”.

As used herein, the term T1 relaxation refers to re-orientation of thenuclear spins in the direction of the primary magnetic field along theaxis Z form the transverse orientation in the plane X0Y perpendicular tothe axis Z. As a result, the global magnetization vector turns by 90degrees and aligns with Z.

As used herein, the term T2 relaxation refers to de-cohering of thespins in the transverse orientation to the primary magnetic field, afterabsorbing the energy of the RF pulse. The resonance nature of the energytransition leads to the same phase of precession for all spins andtherefore the maximal possible transverse magnetization M value. Becauseof passing energy to the environment and field inhomogeneities, theindividual magnetic momenta begin precession in different phases. Theglobal magnetization minimizes, even without considering there-orientation by the T1 mechanism.

As used herein, the term “spin echo” refers to the process of producingthe maximal NMR signal after the RF pulse is turned off, explaining theterm “echo”. The phenomenon arises due to the presence of leading andlagging elements measured relative to the position of the globaltransverse magnetization vector M. Some spins “flip” first and someremain aligned with the primary field producing the lagging elements.The additional energy is supplied producing 180 degrees flip. In thatstronger transverse magnetization environment, the individual spins feelthe local field M as well as B and become synchronized in M not unlikethe iron spin domains in a ferromagnetic after the magnetizing currentis turned off. Since this focusing leads to maximization of the signaland takes place after the excitation energy is not provided any more, itwas termed “echo”.

As used herein, the term “NMR-derived diffusion coefficient” refers tothe result obtained by a process of producing a signal in the detectorcoil by supplying RF energy to produce 90 degree or lower flip angle,destroying the signal by forcefully decohering the transversemagnetization vector by a magnetic gradient pulse, waiting a pre-setinterval of time for the initial phase-labeled population of spins todiffuse away, and refocusing the phase-labeled spins by a combination ofa RF pulse and a magnetic gradient pulse. Phase-labeling means that thatthe spins interacted with a first pair of RF and magnetic pulses willinteract with a second pair, restoring the signal. The process, withvariations, is termed “a sequence”. The difference between the initialand the restored signal can be related to the diffusion coefficient bythe Stejskal and Tanner sequence equations.

FIG. 1 presents the simplified scheme of a typical apparatus for modernNMR analysis. The installation comprises the main magnet, the gradientcoils, the radiofrequency (RF) coils, RF electronics, gradientamplifiers, control electronics, operator console, pulse sequencecomputer and an image reconstruction computer. In operation, the mainmagnet produces a strong magnetic field in the range 0.5-3 Tesla. Thegradient coils superimpose the additional (+) or (−) 0.1 Teslacontributions in the direction transverse to the axis Z (the axis ofrotation inside the main magnet, the axis Z is aligned with thedirection of the main magnetic field). Typically, one gradient iscreated in the direction of the axis X and another in the direction ofY. The extent of the gradient varies along Z. The result of the gradientapplication is partitioning of the space within the combined magneticfields into small elements—voxels, which produce the diverse andvoxel-specific resonance conditions. Based on Fourier seriesdeconvolution it is possible to trace the spin echoes to the individualvoxels (see below) and therefore reconstruct the spatial orientation ofthe factors affecting the signal magnitude (concentrations) andrelaxation times (diffusion coefficients).

The magnetic fields interact with the nuclei possessing uncompensatedspins (odd number of protons or neutrons, counted separately in anucleus). The spins are quantum qualities, but they manifest empiricallyas circular currents running around the nuclei at very high velocity andcreating magnetic momenta, expressed as a product of the current by thearea of the conductive contour (Amper×m²). The momenta are vector valuesand behave at a schematic level analogously to a macroscopic frame withan electric current placed in a magnetic field. Such a frame will alignits normal vector (“magnetic momentum vector”) with the direction of thefield (axis Z), the macroscopic frame will “freeze” perpendicularly tothe field.

The nuclei suitable for NMR comprise, without limitation, the followinglist of isotopes: ²⁷Aluminium, ³⁹Argon, ⁷⁵Arsenic, ¹³⁵Barium, ¹³⁷Barium,⁹Beryllium, ²⁰⁹Bismuth, ¹⁰Boron, ¹¹Boron, ⁷⁹Bromine, ⁸¹Bromine,¹¹¹Cadmium, ¹¹³Cadmium, ⁴³Calcium, ¹³Carbon, ¹³³Cesium, ³⁵Chlorine,³⁷Chlorine, ⁵³Chromium, ⁶³Copper, ⁶⁵Copper, ⁵⁹Cobalt, ²Deuterium,¹⁹Fluorine, ⁶⁹Gallium, ⁷¹Gallium, ⁷³Germanium, ³Helium, ¹Hydrogen,²Hydrogen, ³Hydrogen, ¹¹³Indium, ¹¹⁵Indium, ¹²⁷Iodine, ⁵⁷Iron,⁸³Krypton, ¹³⁸Lanthanum, ¹³⁹Lanthanum, ²⁰⁷Lead, ⁶Lithium, ⁷Lithium,²⁵Magnesium, ⁵⁵Manganese, ¹⁹⁹Mercury, ²⁰¹Mercury, ⁹⁵Molybdenum,⁹⁷Molybdenum, ²¹Neon, ⁶¹Nickel, ¹⁴Nitrogen, ¹⁵Nitrogen, ¹⁸⁷Osmium,¹⁸⁹Osmium, ¹⁷Oxygen, ³¹Phosphorus, ¹⁹⁵Platinum, ³⁹Potassium,⁴⁰Potassium, ⁴¹Potassium, ¹Proton, ¹⁰³Rhodium, ⁸⁵Rubidium, ⁸⁷Rubidium,⁴⁵Scandium, ⁷⁷Selenium, ²⁹Silicon, ¹⁰⁷Silver, ¹⁰⁹Silver, ²³Sodium,⁸⁷Strontium, ³³Sulfur, ¹²³Tellurium, ¹²⁵Tellurium, ¹¹⁵Tin, ¹¹⁷Tin,¹⁹Tin, ⁴⁷Titanium, ⁴⁹Titanium, ³Tritium, ¹⁸³Tungsten, ²³⁵Uranium,⁵⁰Vanadium, ⁵¹Vanadium, ¹²⁹Xenon, ¹³¹Xenon, ⁶⁷Zinc. Of note, deuterium(2H) has the spin value+1, despite having an even number of protons andneutrons. However, the Pauli principle filling of energy levels in allquantum systems apply to each particle type individually; therefore thecombination of a proton and a neutron with parallel spins is not bannedat the same energy level and has a lower energy than an antiparallelcombination, explaining the almost exclusive dominance of the tripletdeuterium vs. a possible singlet state with the antiparallel spins. Bycontrast, two neutrons or two protons at the same energy level must haveantiparallel mutually compensating spins being of the same nature.

The magnetic momenta of the nuclei should ideally align strictly alongthe axis Z, following the direction of the magnetic field B. However,this is impossible due to thermal motion conferring variable quantity oftorque to the spinning nuclei, oriented in the magnetic field.Analogously to a mechanical toy “spinning top” or a gyroscope ingravitational fields, the introduction of the torque by the externalforces alters the orbital momentum of the spinning system.

Newton's second law of motion can be expressed mathematically,

F=m×a   (4)

or force=mass×acceleration. The rotational equivalent for pointparticles may be derived as follows:

L=I×ω  (5)

Wherein I—is the momentum of inertia and ω—is the angular velocity.Thus, the torque τ (i.e. the time derivative of the angular momentum) isgiven as:

τ=(dI/dt)×ω+I×(dω/dt)   (6)

The equation (6) is the rotational analog of Newton's Second Law and thetorque is not always proportional or parallel to the angularacceleration. The external torque introduces a perpendicular componentto the original orbital momentum L_(z) of the body spinning relative tothe axis Z. Under the stationary conditions, this permanentperpendicular component is directed along the tangent to a circulartrajectory of motion accepted by the top of the spinning body. Thecircular motion that establishes after reaching the stationarity phase(after dampening of nutational motion at the non-stationary phase) istermed precession. More than one precession motions are possiblesimultaneously, but for simplicity, only one is assumed in the planeX0Y, normal to the axis Z. Under these simplifying assumptions, theprecession angular velocity is:

$\begin{matrix}{\omega_{p} = {\frac{mgr}{I_{s}\omega_{s}} = \frac{\tau}{I_{s}\omega_{s}}}} & (7) \\{T_{p} = {\frac{4\pi^{2}I_{s}}{{mgrT}_{s}} = \frac{4\pi^{2}I_{s}}{\tau\; T_{s}}}} & (8)\end{matrix}$

Where ω_(p)—is the precession movement angular velocity,T_(p)—precession movement period, I_(s)—is the inertia momentum vs. thespinning axis, ω_(s)—is the angular velocity vs. the spinning axis, τ isthe applied torque. Analogously to mechanical systems, a spinningparticle in a magnetic field experiences a precession movement due toits interactions. When a magnetic moment μ is placed in a magnetic fieldB, it experiences a torque which can be expressed in the form of avector product:

τ=μ×B   (9)

Where μ is the magnetic momentum and B is the magnetic field. For themomentum coinciding with B, the torque is zero but is non-zero for asystem permuted by thermal motion or charge-charge interactions. When atorque is exerted perpendicular to the angular momentum L, it produces achange in angular momentum ΔL which is perpendicular to L, causing it toprecess about the Z-axis. Labeling the precession angle as φ, the effectof the torque can be described as follows:

$\begin{matrix}{\tau = {\frac{\Delta\; L}{\Delta\; t} = {\frac{L\;\sin\mspace{11mu}{\theta\Delta\phi}}{\Delta\; t} = {{{\mu\; B\;\sin\mspace{11mu}\theta}} = {\frac{e}{2m_{e}}{LB}\;\sin\mspace{11mu}\theta}}}}} & (10)\end{matrix}$

The L sin θ is the projection of the original orbital momentum L alignedwith the axis Z onto the perpendicular plane X0Y as a result of torque.ΔL is the change of orbital momentum, and this change is equal to avector difference ΔL between the component L sin θ at the time t₁ andthe same component at the time t₂, resulting due to covering the angleΔϕ in the precession. Also, μ=(e/2m_(e)) L in equation (10) provides alink between torque expressed through magnetic momentum and itsequivalent expressed through angular momentum, both being proportionalvia gyromagnetic ratio (e/2m_(e)).

Considering the definition of torque according to (9) and elementarytransformations produces the final form (11) below. The precessionangular velocity (Larmor frequency) is

$\begin{matrix}{\omega_{Larmor} = {\frac{d\;\phi}{dt} = {\frac{e}{2m_{e}}B}}} & (11)\end{matrix}$

The analysis of (9)-(11) shows that both the original orbital momentum Land the perpendicular component L sin θ induced by the external torqueare canceling in the final expression (11). This canceling is absent inthe mechanical analogy above. The resulting precession frequency isinversely proportional to the inertia m_(e) of the spinning particle,directly proportional to the charge e and the strength of the magneticfield B. The Larmor precession frequency is a characteristic of aparticle in a magnetic field and does not reflect the strength of thepermutations causing deflections from the axis Z (L sin θ is canceled).Yet as a quantum system, a single particle in this minimal energy statewould absorb electromagnetic energy at the same frequency as Larmorprecession.

A mechanical analogy is helpful to illustrate why the resonancecondition is reached at the Larmor frequency. Assuming a heavy spinningtop with an infinite momentum of inertia and periodic impacts by anexternal force, the energy transfer is the most effective when theperiod between the impacts is exactly equal to the period of precession.Moving from a laboratory coordinate system to the one originating in theprecession spinning top leads to the external force arriving atdifferent positions on the precession trajectory if the periods betweenthe impacts and the precession period are different. Over an extendedtimeframe, the positions of impacts will find the opposing and equalcounterparts, thus mutually neutralizing. For equal periods, the impactsarrive always at the same position, leading to the maximizedaccumulation of the transferred energy. The similar principle of theexternal force applied at the inner or natural frequency of oscillationsdefines the better-known conventional resonance.

The individual nuclei oriented along the field B are at the minimalenergy and, when excited by the electromagnetic wave originating in theradiofrequency coils (RF coil), they experience a resonance transitionand “flip” the spins in the opposite direction (against the field). Thisprocess is time-dependent, and with longer exposure to the excitationenergy at the frequency of Larmor precession, the progressively greaterproportion of the individual nuclei changes orientation. If originallythe summary magnetization vector (vector sum of all individual magneticmomenta) was directed along the axis Z with the field, upon “flipping”the summary magnetization vector rotates by 90, 180 or an arbitraryoptimized angle without limitation. FIG. 2 demonstrates the scheme of“flipping” by 90 degrees of the collective magnetization vector.

According to FIG. 2, the initial orientation of the magnetic momenta israndom for the nuclei in the background state (aligned with the fieldwith Boltzmann equilibrium distribution describing the population of theenergy levels). The only net magnetization component is the differencebetween the populations of the background and the excited states, withthe predominant population in the background state (spin polarization).The upper position in FIG. 2 describes this stage. Upon sensing the RFpulse, the system gains energy by the alignment of the magnetic momentain the same phase (lower entropy state, middle position). This phasecoherence is a general property of resonance absorbance. For example,all mechanical pendulums experiencing resonance by the same externalforce are also expected to swing in the same phase. Also, the proportionof the energy levels changes due to the quantum transition to theexcited states, corresponding to flipping some of the spins shown in thelowest position of the figure. The resulting “flipped” collectivemagnetic vector is the original collective magnetic vector turned by 90degrees. In this orientation the net magnetic flux of the sample gyratesin the contour of the RF coils with the Larmor frequency, producing themaximal initial post-flip signal current. The current represents thedecaying oscillations which reflect dephasing and relaxation of thehigh-energy magnetized state after RF pulse is turned off. The signalsare proportional to the square of the primary magnetic field anddirectly proportional to the molar percent of the resonating nuclei. Thesignals can also be deconvoluted as exponential decays with therelaxation times T1 and T2, providing the most useful information aboutthe state of the system. T1 relaxation is the process by which the netmagnetization (M) grows/returns to its initial maximum value (Mo)parallel to Bo in FIG. 2. Synonyms for T1 relaxation includelongitudinal relaxation, thermal relaxation and spin-lattice relaxation.The net magnetization along Z is zero during the action of the RF pulsedue to the M vector flip. The net magnetization returns back to itsBoltzmann distribution maximum by the equation:

M _(t) =M _(max)(1−e ^(−t/T1))   (12)

Where M_(t) is the magnetization at time=t, the time after the 90°pulse, M_(max) is the maximum magnetization at full recovery. This typeof relaxation was termed “spin-lattice” due to lattice or other externalenvironments being the acceptors of the excessive energy in themagnetized material.

As the individual magnetization vectors align with the primary magneticfield, they simultaneously de-cohere since the completely randomorientation of the precessions are more favorable energetically as amore probable state with higher entropy. T2 relaxation is the process bywhich the transverse components of magnetization (M_(xy)) decay ordephase. T2 relaxation is considered to follow first-order kinetics,resulting in a simple exponential decay (like a radioisotope) with timeconstant T2. Thus, T2 is the time required for the transversemagnetization to fall to approximately 37% (1/e) of its initial value.Synonyms for T2 relaxation are transverse relaxation and spin-spinrelaxation (See Bloch F. Nuclear induction. Phys Rev 1946; 70:460-474,1946, incorporated herein by reference in its entirety). T2 relaxationoccurs whenever there is T1 relaxation. Some additional processes alsoexist (such as static local fields and spin “flip-flops”) that cause T2relaxation without affecting T1. T2 relaxation always proceeds at afaster rate than T1 relaxation.

FIG. 3 presents vector diagrams of the individual nuclei explainingacquisition of the 180 “flip angle” in the collective magnetizationstate. Such “flip” angles require twice as much energy absorbed by thesample from the RF pulse, due to either longer duration or higherintensity of the excitation pulsing. If in case of 90 degree “flip”, theindividual magnetization vectors are coherent and 50% are in the excitedstate (producing transverse orientation of the global vector, see FIG.2), in case of 180 degree “flip” 100% of the individual nuclei are inthe excited state. Upon reaching the exact 180 “flip”, no transverseprecession of the global magnetization vector takes place and thereceiving RF coil detects no signal. Only after relaxation begins, theglobal magnetization vector begins to approach the 90 degree “flip”angle when the signal is maximized before it begins to decay.Simultaneously with the T1 relaxation, decoherence and T2 relaxationtakes place all the way from the 180 degree “flip” to the final “flip”state (see below). The 180 degree regime produces more complexrelaxation patterns and communicates qualitatively different informationsupplementing the lower degree regimes. In any real NMR experiment,however, the transverse magnetization decays much faster than would bepredicted by natural atomic and molecular mechanisms; this rate isdenoted T2* (“T2-star”). T2* can be considered an “observed” or“effective” T2, whereas the first T2 can be considered the “natural” or“true” T2 of the formation being imaged. T2* is always less than orequal to T2. T2* results principally from inhomogeneities in the mainmagnetic field. These inhomogeneities may be the result of intrinsicdefects in the magnet itself or from susceptibility-induced fielddistortions produced by the formation or other materials placed withinthe field.

The 180 degree “flips” can be measured not only vs. the originalup-field orientation of the magnetization vector but vs. the initial 90degree “flip” in the sophisticated “spin echo” sequences of the presentinvention. FIG. 4A-F demonstrates such an interpretation of the 180degree “flip”.

Ideally, in each voxel of the apparatus space, all points shouldexperience the same magnetic field and the same synchronized resonanceconditions. However, inhomogeneity in the main magnetic field(“shimming”), local field shielding at some nuclei by valence electrons,and/or different kinetics of re-orientation, create a distribution ofresonance frequencies. Some individual nuclei experience the spin “flip”earlier at lower delivered energy, some proceed together with thepopulation average and some lag behind requiring a greater RF energyinputs to “flip”. As a result, not all components participate information of NMR signal, since some “overshoot” the transverse spinorientation in the plane X0Y that is a prerequisite for signaldetection, while the others remain in the original Z-axis alignedorientation. Thus, the information about the properties of thesecomponents remains unavailable, and the overall analysis becomesincomplete.

The application of saturating levels of RF energy leading to the 180degree “flips” and “spin echo” phenomena addresses these problems (SeeMalcolm H. Levitt; Ray Freeman “NMR population inversion using acomposite pulse”. Journal of Magnetic Resonance, 1979, 33 (2): 473-476;Carr, H. Y.; Purcell, E. M. “Effects of Diffusion on Free Precession inNuclear Magnetic Resonance Experiments”. Physical Review, 1954, 94 (3):630-638; Hahn, E. L. “Spin echoes”. Physical Review, 1950, 80 (4):580-594, incorporated herein by reference in their entireties). Thesequence of vector diagrams illustrating the individual and collectivemagnetization is provided in FIGS. 4A-4F, representing still images fromthe animation of a Hahn echo. The red arrows can be thought of as theindividual spins. Applying the first pulse rotates the spins by 90degrees, producing an equal superposition of spin up and spin down (FIG.2). The spins subsequently “spread out” because each is in a slightlydifferent environment. This spreading out looks like decoherence, but itcan be refocused by a second pulse which rotates the spins by 180degrees. Several simplifications are used in this animation: nodecoherence is included, and each spin experiences perfect pulses duringwhich the environment provides no spreading. The individual spinscontinue focusing after disconnection of RF excitation because thesurviving global transverse magnetization persists for several dozen orhundred milliseconds and the lagging individual elements experience thegreatest focusing force, being perpendicular to the transversemagnetization vector M.

Both T1 and T2 relaxation mechanisms rely on the mobility of individualnuclei. The mobilities in the pores and in the bulk are unequal, withthe more viscous boundary layers imposing faster T2 which dominate theoverall relaxometric signature of the pores (the effects of the fluid inthe bulk on the pore is negligible as compared to the effect of boundarylayer). The interpretation as related to pore geometry is providedbelow.

FIG. 5 presents a qualitative diagram and quantitative data comparing T1and T2 relaxation times in different materials, as a function of thebound vs. free character of the protons and as a function of themolecular masses of each spin-bearing moiety. FIG. 5 illustrates that T1and T2 are the longest for freely moving molecules of smaller mass, butT1 passes through a minimum and begins to increase again formacromolecules, polymers and nanoparticles. T2 is always shorter than T1and is especially short for solid-phase or bound species, alsodemonstrating a much broader dynamic range, thus indicating why T2 isthe parameters of choice for assessing the condition ofhydrocarbon-bearing formation activated by aqueous acidic emulsion. BothT1 and T2 are shorter for hydrocarbons vs. free water.

Rationalizing of these trends is possible recalling that the detectedNMR signal is produced by an ensemble of the flipped spin precessions,rotating in the transverse X0Y plane in the same phase (a necessarysimplification). This assembly stores the energy absorbed from the RFpulse in the two major forms: (a) decreased entropy component bymaintaining phase coherence and (b) orientation of the magnetic momentaperpendicularly to the acting base field B₀. This accumulated excessivepotential energy of the ensemble is transferred to the environment, andthe efficiency of energy transfer determines the duration of therelaxation period.

The precession movements represent a form of oscillations, and if theelements of the environment also oscillate with the same naturalfrequency as the precession frequency, the energy transfer becomes veryefficient, by the same reason as NMR excitation becoming efficient whenthe external RF energy has the same period of oscillations. In case ofrelaxation, it is the oriented molecule in the collective magnetic fieldthat plays the role of the external source and the molecules of theenvironment may or may not provide resonance absorbance, rapidlydepleting the excessive energy of the ensemble. In free water, the ownnatural frequency (“tumbling rate”, rotational degrees of freedom)exceeds the typical Larmor precession frequencies by orders ofmagnitude. Thus, the energy transfer is ineffective, and the relaxationtimes can be 3-5 seconds long for both T1 and T2. Hydrocarbons representheavier molecules than water with slower “tumbling rate” producing moreefficient energy transfer and dissipation for both T1 and T2. Polymersare too slow, and once they acquire transverse magnetization, thepassage of energy becomes inefficient due to the “tumbling rate” (ownoscillation frequency) of the entire molecules becoming much slower thanthe precession rate of the nuclei within them. Thus, T1 increases. Dueto a very larger number of degrees of freedom, maintaining precessionphase coherence in large molecules requires a very large entropyconstraint which is not a probable state. Thus, the magnetizations inthese particles decohere rapidly even if the entire molecules do notre-orient rapidly along Z-axis, leading to the shortened T2 at longerT1. In ice, re-orientation of the magnetization vector in parallel to Zaxis is hindered by the participation of the water molecules in closeorder crystal structure, leading to the “tumbling rates” much slowerthan the nuclear precession frequency and therefore high T1. An icecrystallite can be considered as a polymer-like structure in whichmaintaining magnetization phase coherence would require large entropyconstrain and therefore decohering is energetically favorable, producingshort T2 and long T1.

Thus, NMR measurements are sensitive to the mobility of species, to thenumber of individual species, to the extent of participation in largestructures, free or bound character, freely migrating or covalentlybound hydrogen status. Sensitivity of NMR to so many parameters (seebelow) requires development of back-up alternatives. The Measuring WhileDrilling (MWD) alternatives to sample extraction, delivery to alaboratory and application of the results to the process upon completionof the laboratory study are advantageous (See Prammer, M. G., Drack, E.,Goodman, G. et al. The Magnetic-Resonance While-Drilling Tool: Theoryand Operation. SPE Res Eval & Eng, 2001, 4 (4): 270-275. SPE-72495-PA;Appel, M., Radcliffe, N. J., Aadireddy, P. et al. Nuclear MagneticResonance While Drilling in the U.K. Southern North Sea. Presented atthe SPE Annual Technical Conference and Exhibition, San Antonio, Tex.,USA, 2002, 29 Sep.-2 Oct. SPE-77395-PA; Morley, J., Heidler, R.,Horkowitz, J. et al. Field Testing of a New Nuclear Magnetic ResonanceLogging-While-Drilling Tool. Presented at the SPE Annual TechnicalConference and Exhibition, San Antonio, Tex., USA, 2002, 29 Sep.-2 Oct.SPE-77477-MS; Poitzsch, M., Scheibal, J. R., Hashem, M. et al.Applications of a New Magnetic Resonance Logging-While-Drilling Tool ina Gulf of Mexico Deepwater Development Project. Presented at the SPWLA43rd Annual Logging Symposium, Oiso, Japan, 2002, 2-5 Jun.SPWLA-2002-EEE; Heidler, R., Morriss, C., and Hoshun, R. 2003. Designand Implementation of a New Magnetic Resonance Tool for the WhileDrilling Environment. Presented at the SPWLA 44th Annual LoggingSymposium, Galveston, Tex., USA, 2002, 22-25 Jun. SPWLA-2003-BBB;Borghi, M., Porrera, F., Lyne, A. et al. Magnetic Resonance WhileDrilling Streamlines Reservoir Evaluation. Presented at the SPWLA 46thAnnual Logging Symposium, New Orleans, 2005, 26-29 Jun. Paper 2005-HHH).

In one embodiment, the method relies on the commercially availabledrilling assemblies incorporating NMR apparatus among other sensors forproviding the real-time multifactorial feedback. Non-limiting examplesinclude ProVision Plus (magnetic resonance while drilling apparatus bySchlumberger). The apparatus and the software acquires the T2 dataderived from pore size and fluid properties within several seconds, theT2 data can be further processed to quantify bound- and free-fluidvolume and capillary-bound water. MR signal decay data helps ascertainpermeability, producible porosity, and irreducible water saturation aswell as capillary pressure curves, hydrocarbon identification, andfacies analysis. A real-time permeability log enables predictingproduction rates to optimize completion and stimulation programs.

Analogously, Halliburton Sperry Drilling apparatus and software offerslogging-while-drilling (LWD) nuclear magnetic resonance (NMR)source-less porosity solutions to help gain insight into the keypetrophysical properties of the reservoir through amineralogy-independent porosity assessment. The apparatus (MRIL®-WD™Magnetic Resonance Imaging Logging-While-Drilling Sensor) determine thetotal porosity of a reservoir, movable fluid volume, capillary boundfluid volume, and micro-porosity—essential information to allow a userto determine which fluids produce hydrocarbons upon stimulation. Bycalculating a qualitative estimate of permeability, the user identifieswhich reservoir intervals have greater flow, and can better target theoptimal spots for increased production.

Other providers of the small-diameter borehole-adapted NMR apparatusescomprise Mount Sopris Instruments, Baker Hughes Incorporated tools,Dart, Javelin, and Javelin Wireline tools by VISTA-CLARA INC withoutlimitation. These and the additional MRWD (magnetic resonance whiledrilling) designs are disclosed in U.S. Pat. Nos. 10,197,698,10,401,313, 10,473,600, 10,295,627, 10,338,267, US20190033483, U.S. Pat.No. 10,191,178 incorporated herein by reference in entirety.

In a preferred embodiment, the commercial, experimental and customizeddownhole NMR apparatuses can support diffusion NMR and can produce themagnetic gradient pulses. Diffusion NMR methods resolve differentcompounds spectroscopically in a mixture based on their differingdiffusion coefficients, depending on the size and shape of themolecules. Diffusion NMR may be used to resolve otherwise intractablespectra of mixtures, or it may be used to determine the size ofmolecules and aggregates, determining the degree of polymerization, sizeof a solvation shell or other microscopic structure. The spectraproduced resemble chromatograms is some respects while also providingNMR information that can be used for the assignment of individualcomponents.

FIGS. 6A-D illustrate the principles of diffusion NMR with a magneticpulse gradient. As shown in FIG. 6A, the magnetization vector is already90 degrees “flipped” by the RF pulse, producing a detectable signal inthe transverse detector coil. The precession movement of the nuclei issynchronized in the resonance transition. The term “dephasingpulse-field gradient” refers to the gradient of the magnetic fieldgenerated by the gradient magnetic coils (FIG. 1), to be differentiatedfrom the RF pulse generated by RF coils. Imposition of different Bvalues in the analyzed volume by the magnetic gradient pulse leads todifferent precession frequencies and therefore decoheres the initialmagnetized environment post 90 degree “flip”. No signal can be detected,since the global magnetization vector becomes zero. “Refocusing pulsefield gradient” re-orients the nuclei that were in phase after the 90degree “flip” and restores the signal in the detector, however, thediffusion of the species out of the volume that communicates with thedetector coil diminishes the restored signal. Thus, the differencebetween signals before and after the sequence cycle provides the basisfor estimating the diffusion coefficients. FIG. 6B illustrates therefocusing aspect. After a period of Δ/2 a 180° radiofrequency pulseinverts the dispersed magnetization such that after a period of Δ themagnetization is the negative of what it was following the gradientpulse. At this point, a second gradient pulse is applied to refocus thesignal. The refocusing develops due to two contributions: one is the 180degree “flip” counted vs. the initial 90 degree “flip” as typical inecho sequences and accomplished by an RF pulse (blue upper signals inFIG. 6B). The magnetization focusing by this mechanism was disclosedabove. The second mechanism is the reversal of the magnetic pulse (thesecond pulse of the magnetic gradient), shown in red in FIG. 6B. Thephase diagram in FIG. 6C illustrates the interaction between the180-degree RF pulse and the second magnetic gradient pulse. The FIG. 6Cclarifies that the individual magnetic momenta of the nuclei that lagbehind the neighbors or overtake them in terms of precession movementreturn to the previous position after the 180 degree RF and the secondmagnetic gradient pulse (inverted vs. the first gradient pulse),restoring the coherence and allowing to observe the signal. The PGSEsequence leads to identification and diffusion coefficient measurementfor the diverse species.

In PGSE in the case of self-diffusion of molecules, the normalizedsignal amplitude E/E0 decays as a Gaussian curve with increasingmagnetic gradient pulse amplitude G:

$\begin{matrix}{\frac{S({TE})}{S_{0}} = {\exp\left\lbrack {{- \gamma^{2}}G^{2}{\delta^{2}\left( {\Delta - \frac{\delta}{3}} \right)}D} \right\rbrack}} & (13)\end{matrix}$

where S₀ is the signal intensity without the diffusion weighting, S (TE)is the signal with the magnetic field gradient, γ is the gyromagneticratio, G is the strength of the gradient pulse, δ is the duration of thepulse, Δ is the time between the two pulses, and finally, D is thediffusion coefficient.

FIG. 6D shows a stacked experiment, where the strength of the residualsignal S(TE) is plotted as a function of the magnetic field gradient G.The FIG. 6D illustrates how the resonance signal emerges at theresonance RF frequency f1 and how the signal decreases in the serialmeasurements as a function of dephasing magnetic gradient pulse.

In a preferred embodiment, the apparatus and software support partialflip angles with more rapid accumulation of multiple data points toproduce logarithmic plots. The plots are linear in the coordinates [lnS(TE)/S0=k G²] where the coefficient k includes the diffusioncoefficient and the cycle pulse parameters. The rate of data pointaccumulation is important to increase the signal-to-noise ratio, whichmay be high in the NMR method of diffusion measurement due to highpropagated error. In this embodiment, the partial flip angles range from10 to 85 degrees without reaching the 90 degrees, which is achievable bysofter RF pulses. The increased rate of data acquisition is achievableunder these conditions due to shorter periods of magnetization energyaccumulation and relaxation. In addition, partial flip angles alleviatethe limitations on sensitivity that arise from the quantum-mechanicalnature of the phenomenon. For quantum states separated by energyequivalent to radio frequencies, thermal energy from the environmentcauses the populations of the states to be close to equal. Sinceincoming radiation is equally likely to cause stimulated emission (atransition from the upper to the lower state) as absorption, the NMReffect depends on an excess of nuclei in the lower states. Severalfactors can reduce sensitivity, including increasing temperature, whichevens out the population of states.

Conversely, low-temperature NMR can sometimes yield better results thanroom-temperature NMR, providing the sample remains liquid. Saturation ofthe sample with energy applied at the resonant radiofrequency (completeflip angles, 90 degrees or above) is another sensitivity-reducingfactor. This manifests in both constant wave (CW) and pulsed NMR. In thefirst case (CW) this happens by using too much continuous power thatkeeps the upper spin levels completely populated. In the second case(pulsed), each pulse (that is at least a 90° pulse) leaves the samplesaturated, and four to five times the (longitudinal) relaxation time(5T1) must pass before the next pulse or pulse sequence can be applied.For single pulse experiments, shorter RF pulses that tip themagnetization by less than 90° can be used, which loses some intensityof the signal, but allows for shorter recycle delays. The optimum “flip”angle is called an Ernst angle. The relaxation times for the protons infree water are relatively short, but in the drilling conditions,nanoparticles and hydrocarbon-based aggregates contribute slowlyrelaxing components, and the use of partial angles can be advantageous.

In another preferred embodiment, multiple diffusion coefficients aresimultaneously measured in combination by pulsed NMR, with the deviceand software supportive of data generation and analysis. FIG. 7 presentsDiffusion-Ordering Spectroscopy NMR (DOSY). The figure presents a 2Dplot with the abscissa being the chemical shifts and the ordinate beingthe magnitude of the magnetic gradient (See equation 13). The chemicalspecies are identified by the shifts, and the decay of the signal is afunction of magnetic gradient pulse strength G is plotted for eachchemical shift. As far as data processing of raw PFG-NMR spectra isconcerned, the goal is to transform the N×M data matrix S into an N×Rmatrix (2D DOSY spectrum) as shown in FIGS. 8A and 8B. The horizontalaxis of the DOSY map D is identical to that of S and encodes thechemical shift of the nucleus observed (generally 1H). The verticaldimension, however, encodes the diffusion coefficient D. In the idealcase of non-overlapping component lines and no chemical exchange, the 2Dpeaks align themselves along horizontal lines, each corresponding to onesample component (molecule). The horizontal cut along such a line shouldshow that the component's ‘normal’ spectrum of chemical shifts. Verticalcuts show the diffusion peaks at positions defining the correspondingdiffusion constants. The mapping S=>D is called the DOSY transformation.This transformation is, unfortunately, far from straightforward.Practical implementations of the procedure include mono andbiexponential fitting, Maximum Entropy, and multivariate methods such asDECRA ‘Speedy Component Resolution’ (See M. Nilsson and Gareth A. Morrisin Anal. Chem., 2008, 80, 3777-3782 incorporated herein by reference inentirety) as an improved variation of the Component Resolved (CORE)method (J. Phys. Chem, 1996, 100, 8180, incorporated herein byreference) providing a multivariate-based that shows an advantageousperformance of the algorithm.

In another preferred embodiment, the apparatus and software support theadditional sequences suitable for the diffusion coefficientmeasurements: the Hahn stimulated echo (STE) with pulsed field gradients(FIG. 9). The sequence is analogous to PGSE but differs by providing two90 degrees RF pulses in place of one 180-degree RF pulse in PGSE and byinsertion of an additional “crusher” magnetic gradient pulse step. TheLED (longitudinal encode decode) pulse sequence used in the NMRdiffusion experiments is shown in FIG. 10. The radiofrequency pulses areshown as filled vertical rectangles with the flip angles denoted aboveeach pulse. The magnetic field gradient pulses are shown as hatchedrectangles, and the data acquisition is indicated with a verticallyhatched triangle. Bipolar gradient longitudinal encode-decode pulsesequence (BPP-LED) is shown in FIG. 11. The BPPLED method cancels theadverse effects of eddy currents using two gradient pulses withidentical areas but different polarities. Other embodiments comprise,respectively, the gradient compensated stimulated spin-echo pulsesequences (GCSTE), the double stimulated echo sequence (DSTE), theSTE-INEPT pulse sequences for heteronuclear detected DOSY with coherencetransfer, shuttle based fringe field 2D-DOSY, and the Alternating PulsedField Gradient Stimulated Echo Nuclear Magnetic Resonance (APGSTE)sequence (FIG. 12) without limiting. The APGSTE sequence is especiallypreferred for analyzing anisotropic diffusional systems, such ashydrocarbon-bearing formations with anisotropic porosity and tortuositydistributions. The sequence comprises focusing and de-coherence by theseries of magnetic gradient pulses in 3 dimensions, explaining itsunique suitability for more realistic modeling of diffusionalcoefficient tensors. All sequences lead to processing and datacollection/transform by the DOSY methodology, with the Tanner andStejskal equation modified for each specific sequence (See Jan Hrabe,Gurjinder Kaur, and David N. Guilfoyle, “Principles and limitations ofNMR diffusion measurements” in J Med Phys., 2007 January-March; 32(1):34-42; Davy Sinnaeve, “The Stejskal-Tanner Equation Generalized for AnyGradient Shape—An Overview of Most Pulse Sequences Measuring FreeDiffusion” in Concepts in Magnetic Resonance Part A, 2012, Vol. 40A(2)39-65, incorporated herein by reference in entirety).

In the most preferred embodiments, the in-situ “logging-as-drilling” NMRdata are applied to the calculation or estimation of porosity byutilizing relaxation times and tortuosity by utilizing measuredrestricted diffusion.

The physical basis for using restricted diffusion for assessing poregeometry results from the following: diffusion coefficient measures thenumber of elementary “leaps” between local potential energy minima. Thisnumber of “leaps” needs to increase to carry a labeled particle to thetarget destination, if the trajectory is tortuous as compared to acontrol where the trajectory is linear. The spin precession coherence inpulsed NMR measurements is a variant of labeling (FIG. 6C). The distancethe spin-labeled particle can escape from the previous position isproportional to time-dependent loss of NMR signal (see the discussion ofFIGS. 6 and 7) and forms the basis for diffusion coefficientmeasurement. In a tortuous environment this distance is reduced becausemore elementary leaps are required to accomplish the same displacement,or for the same period of NMR sequence time—a lesser displacement isachieved per the same number of elementary leaps. The differencesbetween restricted and bulk diffusion develop as a function of time.Over shorter time intervals, the number of elementary leaps outside thepores and in the pore environments is comparable. However, over a longerinterval, the diffusing particle experiences collisions with the porewalls, while the motion outside the pores is unhindered, increasing theratio. The real effect of diffusional restriction needs to be observedin the asymptotic range of this time-dependence, forcing certain minimallengths of the NMR sequence duration (See FIGS. 14-17 in the Examples).

This time-dependence of the restricted diffusion ratios is variable forthe pores of different radii and can be used to evaluate pore sizedistribution, together with the T2 relaxation data considered below.Both methods are based on different physical principles and thusrepresent the independent predictors of other economically importantproperties, such as rock permeability, wettability and mechanicalstrength. The time-dependence of restricted diffusion can be alsoanisotropic to a variable degree in different brackets of pore sizes.

Another complexity is the heterogeneity of the flow through the pores.The assumption is that the formation is flooded and that the diffusing1H nuclei belong to water. Only when the 1H nuclei outside of the poresand within the pores demonstrate comparable inherent mobility (frequencyand length of the elementary leaps), the observed differences areattributable to restriction alone. The water may be diffusion-restricted(from the NMR viewpoint) not only by the geometry of the pores but byforming microemulsions in the hydrocarbon flow or being bound on thepore surfaces. The anisotropy conclusion needs to be verified comparingdifferent fractions of the NMR signal in terms of relaxation times andchemical shifts. For example, the pore anisotropy conclusion reachedusing water chemical shifts should agree with the conclusions reachedusing hydrocarbon chemical shifts. Typically, in a flooded well there isenough water saturation in the pores that can be filtered by relaxationtimes and chemical shifts, matching that in the bulk water outside ofthe porous media. Thus, the assumption is that the 1H NMR studies waterand the primary data are filtered for the non-emulsified pore-dwellingwater contributions. Mud fluids used to cool the rigs and remove thedrilling cuttings ensure that the proximal surface 0-5 mm deep into theborehole wall satisfies the criteria above.

Tortuosity has different definitions depending on the method ofmeasurement (B. Ghanbarian, A. G. Hunt, R. P. Ewing, and M. Sahimi,“Tortuosity in Porous Media: A Critical Review,” Soil Sci. Soc. Am. J.,vol. 77, no. 5, p. 1461, 2013—incorporated herein by reference in itsentirety). The definition of this disclosure is “diffusive tortuosity,τ_(d)” which is defined as the ratio of the free self-diffusioncoefficient of bulk fluid, D₀, to the restricted diffusion coefficientof fluid in the porous media, D_(R):

τ_(d) =D ₀ /D _(R)   (14)

One of the most accurate non-invasive techniques to measure diffusioncoefficients in porous media is NMR pulsed field gradient (NMR PFG, seeFIGS. 4-12 and discussion herein). Recent studies (See K. Yang et al.,“Quantitative Tortuosity Measurements of

Carbonate Rocks Using Pulsed Field Gradient NMR,” Transp. Porous Media,September 2019, incorporated herein by reference in entirety) showedthat NMR PFG can be utilized to measure τ_(d) in porous rocks since suchparameter is essential for enhanced gas recovery studies. The magneticfield gradient in NMR PFG can be applied in different directions, andtherefore anisotropy in diffusion coefficient can be evaluated. Based onthis concept, NMR PGF is used in the field of physical chemistry andmaterial sciences to evaluate diffusion anisotropy in plant tissues,liquid crystals and battery electrodes (S. Engelke, L. E. Marbella, N.M. Trease, M. De Volder, and C. P. Grey, “Three-dimensional pulsed fieldgradient NMR measurements of self-diffusion in anisotropic materials forenergy storage applications,” Phys. Chem. Chem. Phys., vol. 21, no. 8,pp. 4538-4546, January 2019, Y. Qiao, P. Galvosas, and P. T. Callaghan,“Diffusion Correlation NMR Spectroscopic Study of Anisotropic Diffusionof Water in Plant Tissues,” Biophys. J., vol. 89, no. 4, pp. 2899-2905,October 2005, I. Furo and S. V. Dvinskikh, “NMR methods applied toanisotropic diffusion,” Magn. Reson. Chem., vol. 40, no. 13, pp. S3-S14,December 2002).

In a preferred embodiment of the present disclosure, NMR PFG is used toevaluate permeability anisotropy in hydrocarbon-reservoir rock.Evaluating τ_(d) in both vertical and horizontal directions (withrespect to the layering direction) can quantitate the permeabilityanisotropy in hydrocarbon reservoirs and their later use for large-scalemodeling of borehole productivity. FIG. 13 shows a representation of howthe NMR gradient is utilized to measure the anisotropic characteristicof the rock samples. If the z-direction gradient is applied, themovement of the molecules is in the vertical direction of the sample,which gives an indication of the vertical permeability of the rocksample. On the other hand, if the x-direction gradient is applied themovement of the molecules is in the horizontal (radial) direction.

In a preferred embodiment, a stand-alone NMR methodology (no supportingmethods) determines tortuosity, porosity, pore size distributions andanisotropy of tortuosity and porosity. The advantage of a stand-alonemethod is simplicity of the downhole sensor assembly relying on a singleapparatus to collect the log data suitable for further machine learningprocessing. The stand-alone method is less reliable than a methodutilizing multiple principles, but may be cheaper if applied in the samebasin for the same mineralogical composition.

In a stand-alone NMR method, porosity is determined relaxometrically bydeconvoluting the information produced at post-RF pulse stages in themagnetic gradient pulse sequences. FIGS. 9-12 all describe the intervalsof time between the RF and magnetic pulses when the magnetized mediarelaxes via the T₂ mechanism. Thus, two measurements—of restricteddiffusion coefficients and of relaxation patterns—are performed by theanalysis of a single sequence cycle. The reciprocal of relaxation timein a particular region is proportional to the magnetization energydissipated by the region of interest (See Valfouskaya, A.; Adler, P. M.;Thovert, J. F.; Fleury, M. (2005), “Nuclear magnetic resonance diffusionwith surface relaxation in porous media”, Journal of Colloid andInterface Science, 295 (1): 188-201; Brownstein, K. R.; Tarr, C. E.(1979), “Importance of classical diffusion in NMR studies of water inbiological cells”, Physical Review A, 19 (6): 2446; incorporated hereinby reference in entirety). The balance expression for one pore:

$\begin{matrix}{\frac{1}{T_{i}} = {{\left( {1 - \frac{\delta\; S}{V}} \right)\frac{1}{T_{ib}}} + {\frac{\delta\; S}{V}\frac{1}{T_{is}}} + {D\frac{\left( {\gamma\;{Gt}_{E}} \right)^{2}}{12}}}} & (15)\end{matrix}$

Where:

δ is the thickness of the surface area of the bound water on the pore'swalls,

S is the surface area of the pores,

V is the pore volume,

T_(ib) is the relaxation time in the bulk volume,

T_(is) is the relaxation time for the surface,

γ is the gyromagnetic ratio,

G is the magnetic field gradient (assumed to be constant),

t_(E) is the time between echoes and

D is the self-diffusion coefficient of the fluid.

The amplitude of NMR signal is proportional to the number of 1H nuclei.In the samples with greater porosities, the strength of the signal isproportionally higher, and porosity fraction can be estimated measuringthe signal from the porous material and bulk water. The bulk V of a poreexhibits a single exponential decay. The nuclei close to the pore wallsurface S exhibits faster T₂ than the bulk, due to higher viscosity(“ice-likeness”) of the bound water. Since the pores are assumed to beunconnected and small, the diffusivity loss of energy and relaxation inthe pore bulk are neglected.

With T_(is)<<T_(ib), the expression (15) transforms:

$\begin{matrix}{\frac{1}{T_{2}} = \frac{\rho\; S}{V}} & (16) \\{{M(t)} = {M_{0}e^{{- t}/T_{2}}}} & (17)\end{matrix}$

Where:

S/V is the surface to volume ratio of a single pore in the givendirection,

ρ is the surface relaxation strength (the intensity of magnetizationenergy dissipation due to interaction with the surface-bound water),

M(t), M₀ are, respectively, the current and the initial magnetization inthe given pore.

Considering the distribution of the pores, the expression becomes:

$\begin{matrix}{{M(t)} = {M_{0}{\sum\limits_{i = 1}^{n}{a_{i}e^{{- t}/T_{2}}}}}} & (18)\end{matrix}$

Where a_(i) are the volume fractions of the pores with the surface tovolume ratios (18), defining the values T₂. The M(t) values in (18) aregrouped based on the pore radii and finite numbers of pore diametercategories fit the experimental time-dependent values M(t).

To summarize, the relationships (14)-(18) enable determining theporosity (the fraction of the material filled with fluid (ϵ)), the sizedistribution of the pores and tortuosity of each path, all measurementsperformed in several directions by NMR alone. These data allow modelingof viscous flow permeability across a unit volume of the studied porousmaterial. This modeling is applicable only to continuous fluid flows:natural gas, rich oil deposits uniformly wetting the rock pore surface,processing fluids. According to the Darcy equation:

$\begin{matrix}{{\frac{\Delta\; p}{L} = {\frac{128}{\pi} \cdot \frac{\mu\; Q}{D_{c}^{4}}}},} & (19)\end{matrix}$

Where:

-   -   ΔP/L is the full gradient of lithostatic pressure in the        direction of hydrocarbon flow-path,    -   D_(c) is the pore diameter,    -   μ is the dynamic viscosity of the hydrocarbon, migrating toward        the borehole and/or induced fractures,    -   Q is the volume flow [m³/sec] per the unit area.

The equation (22) incorporates the NMR-derived data:

ΔP/L=(τ/ϵ)×(128×Q×μ/π)×[Σa ₁×(D _(c))⁻⁴]  (20)

Where:

ϵ is the porosity of the hydrocarbon reservoir [dimensionless fraction],determined by measuring 1H magnetization M(t) in a porous sample andcomparing with the bulk fluid reference, and scaling the reference datato the same volume as in the sample,

τ is the diffusional tortuosity determined by the restricted diffusionanalysis,

a₁ are the volume fractions of the pore distribution, determined by NMRrelaxometry. According to (20), per the same volume flow Q, a higherpressure gradient is required if the sample displays high tortuosity,low porosity, and a high fraction of micropores. Conversely, lower flowrate is expected at the same pressure gradient under the conditions asabove.

In a preferred embodiment, the equation (20) is a part of acomputational process for estimating the productivity of a drillingproject and relating the calculated production yield to the costs (thecontinuity limitation was already discussed). The computational processbegins with modeling of the hydrocarbon reservoir, assessing itsdimensions as well as the percentage of hydrocarbon entrapped in thepores and fractures of the formation. Potential hydrocarbon depositsalter the density of the sub-terrain layers within 0.001-0.01% of g,enabling gravity surveys. Gravimeters used in geophysical surveys havean accuracy of about 0.01 milligal or mgal (1 milligal=0.001 centimeterper second per second). These devices detect differences in the Earth'sgravitational field as small as one part in 100,000,000 (See FredAminzadeh, Shivaji N. Dasgupta, in Developments in Petroleum Science,2013, incorporated herein by reference in entirety). Gravity modeling issupplemented by measuring magnetic anomalies, known to correlate withhydrocarbon depositions even if the nature of these anomalies is notfully understood (See Donovan T J, Forgey R L, Roberts A A. Aeromagneticdetection of diagenetic magnetite over oil fields: geologic notes. AAPGBulletin. 1979 Feb. 1; 63(2):245-8, Reynolds R L, Fishman N S, Hudson MR. Sources of aeromagnetic anomalies over Cement oil field (Oklahoma),Simpson oil field (Alaska), and the Wyoming-Idaho-Utah thrust belt.Geophysics. 1991 May; 56(5):606-17; Eventov L. The nature andinterpretation of geophysical and geochemical anomalies over oil and gasfields. The Leading Edge. 2000 May; 19(5):488-90, each incorporatedherein by reference in its entirety). The observation area ofgravitational and magnetic anomalies produces an estimate of horizontaldimensions for the deposit, while the intensity allows modeling thedepth. At the next step, seismological studies are conducted bygenerating explosive shockwaves and monitoring reflection of the soundfrom the boundaries of the strata with different compressibility anddensity. The time lag before the detection of the reflected waveindicates the depth. Launching the shockwaves at different anglesproduces 3D seismological data, enabling tomographic analysis withimproved feature resolution (See Tape C, Liu Q, Maggi A, Tromp J.Adjoint tomography of the southern California crust. Science. 2009 Aug.21; 325(5943):988-92; Tselentis G A, Martakis N, Paraskevopoulos P, LoisA. High-resolution passive seismic tomography for 3D velocity, Poisson'sratio v, and P-wave quality QP in the Delvina hydrocarbon field,southern Albania. Geophysics. 2011 May; 76(3):B89-112; Martakis N,Landtech Enterprises Sa, Tselentis A. Parakevas Paraskevopoulos, Dept.of Seismology, University of Patras, Greece. High Resolution PassiveSeismic Tomography—a NEW Exploration Tool for Hydrocarbon Investigation,Recent Results from a Successful Case History in Albania; incorporatedherein by reference in entirety). Features of interest (known as“leads”) are subjected to more detailed seismic surveys using theprocess of depth conversion to create a profile of the substructure.Finally, when a prospect has been identified and evaluated and passesthe selection criteria, an exploration well is drilled to determine thepresence or absence of oil or gas conclusively. The drilling stepestablishes the presence of the hydrocarbon deposit, may includemineralogical sampling, optionally assessment of porosity, wettabilityand/or hydrocarbon saturation of the rocks, optionally hydrocarbonsampling and chemical analysis. The porosity parameters are preferablymeasured at the explorative drilling stage to inform business decisions.

The decision to terminate is likely to follow adetermination/calculation of low formation permeability and lowhydrocarbon volume percentage. At certain percentages, the hydrocarbonwill not migrate to the development well. At certain wettability values,the formation will retain a non-moving hydrocarbon and repel theprocessing fluids, complicating recovery. Additional optionalconsiderations include the asphaltene content of the hydrocarbon, theacidity of the hydrocarbon, and the presence of catalytic poisonscomplicating future refining of the hydrocarbon. The decision tocontinue preferably includes incorporating the reservoir dimensions,horizontal and vertical distribution of the sub-zones, and permeability.The latter includes anisotropic porosity, pore distribution andtortuosity as shown in (20).

Typical threshold values for permeability, wettability index andhydrocarbon content can vary depending on reservoir pressure and rocktype. Lower threshold values (i.e., a non-producible threshold) thatdefine from a producible boundary are as follows:

Permeability: 0.01-10 mD, 0.1-5 mD, 0.5-1 mD or about 1-3 mD (the higherthe permeability, the easier the production); hydrocarbon percentage offormation rock based on weight: 10-50%, 15-40%, 20-30% or about 25% withhigher values preferred; wettability index: recovery is difficult forstrongly oil wet rocks (wettability index from −0.4 to −1, −0.6 to 0.8or about −0.6); while production is relatively easier in wettabilityranges of from −0.3 to 1, −0.1 to 1, or 0 to 0.5.

With the permeability values calculated/determined, and the overallsuitability of the deposit affirmed, a drilling location and depth canbe mapped on the model. Different sub-zones of the modeled hydrocarbonreservoir can be located at varying distances, depths and communicatewith the borehole via diverse paths with generally unequal hydraulicresistivity. The estimates of horizontal and vertical permeability allowcomputing a profile of driving force (ΔP) and hydraulic resistivity fromall sub-zones to the proposed downhole position. For more precision, thesubzones are divided in volume units—voxels. The extent of fracturing isalso included in the model (to extend the surface of hydrocarbonrelease), with the tradeoff of weakening the boreholes which requireadding hydraulic pressure to stabilize the flood-work. The addedhydrostatic and hydraulic pressure opposes lithostatic pressure anddiminishes the driving force of the process. With all factors accountedfor, a daily yield is computed together with the expected lifespan andthe capital/amortization/exploitation costs of the well. The product ofmarket price of the hydrocarbon by the yield over the borehole lifespanproduces the revenue stream, while the costs are taken from thereference data. Multiple locations, well branching schemes andfracturing regimes are computationally tested until a set of the bestresults is identified. The choice of a permeability model is essentialfor the outcome of economic modeling.

Considering the limitations of NMR-only data under the realisticconditions of a processed well, supporting methods are incorporated inthe analysis. In a preferred non-limiting embodiment, NMR-only data aresupplemented by resistivity log data, following Archie's equation. Inpetrophysics, Archie's law relates the in-situ electrical conductivityof sedimentary rock to its porosity and brine saturation:

$\begin{matrix}{C_{t} = {\frac{1}{a}C_{w}\phi^{m}S_{w}^{n}}} & (21)\end{matrix}$

Here,

-   ϕ—is the porosity,-   C_(t)—the electrical conductivity of the fluid-saturated rock,-   C_(w)—represents the electrical conductivity of the brine, in the    same volume as the rock,-   S_(w)—is the brine saturation (fraction of the sample's volume    occupied by the brine, <1),-   m—is the cementation exponent of the rock (usually in the range    1.8-2.0 for sandstones, 1.7-4.1 for carbonates),-   n—is the saturation exponent (usually close to 2 for sandstones and    carbonates) and-   a—is the tortuosity factor.    Reformulated for electrical resistivity, the equation reads:

R _(t) =aϕ ^(−m) S _(w) ^(−n) R _(w)   (22)

-   R_(t)—is the fluid-saturated rock resistivity, and-   R_(w)—is the brine resistivity.    The factor

$\begin{matrix}{F = {\frac{a}{\phi^{m}} = \frac{R_{o}}{R_{w}}}} & (23)\end{matrix}$

is also called the formation factor, where

-   R_(o)—is the resistivity of the rock volume filled with only water    (S_(w)=1, no solid)-   R_(w)—is the resistivity of the porous rock with the enclosed water    The factor

I=R _(t) /R _(o) =S _(w) ^(−n)

is also called the resistivity index.

The cementation factor m estimates how the pore structure andconnectivity increase the resistivity, as sandstone or carbonate itselfis non-conductive. If porosity were to be modeled as parallel channels,a cross-section area average of the rock's resistivity would yieldporosity dependence equivalent to a cementation exponent of 1. However,the tortuosity of the rock increases this to a higher number than 1.This relates the cementation exponent to the permeability of the rock;increasing permeability decreases the cementation exponent.

The saturation exponent typically approaches 2. The saturation exponentmodels the dependence of conductivity on the presence of non-conductivefluid (hydrocarbons) in the pore-space and is related to the wettabilityof the rock. Water-wet rocks will, for low water saturation values,maintain a continuous film along the pore walls making the rockconductive. Oil-wet rocks will have discontinuous droplets of waterwithin the pore space, making the rock less conductive.

The tortuosity factor a in Archie's law is not the same as diffusionaltortuosity, it is a mathematical constant, fitting the form to theobserved data, thus a can be 0.5-0.6, while real tortuosity isalways >1.0. However, the formation factor F is the resistivityequivalent of NMR-derived tortuosity.

Archies' law is empirical, and the constants n and m are derived on theactual samples from the oil reservoirs. Different mathematicaltreatments are known in the art (See Archie, G. E. (1942). “Theelectrical resistivity log as an aid in determining some reservoircharacteristics”. Petroleum Transactions of AIME. 146: 54-62; Archie, G.E. (1947). “Electrical resistivity an aid in core-analysisinterpretation”. American Association of Petroleum Geologists Bulletin.31 (2): 350-366; Archie, G. E. (1950). “Introduction to petrophysics ofreservoir rocks”. American Association of Petroleum Geologists Bulletin.34 (5): 943-961; Archie, G. E. (1952). “Classification of carbonatereservoir rocks and petrophysical considerations”. American Associationof Petroleum Geologists Bulletin. 36 (2): 278-298; Etris E L, BrumfieldD, Ehrlich R, Crabtree S J. Petrographic insights into the relevance ofArchie's equation: formation factor without “m” and “a. In SPWLA 30thAnnual Logging Symposium 1989 Jan. 1. Society of Petrophysicists andWell-Log Analysts; Aldoleimi A M, Berta D. A new technique for derivingthe resistivity parameters of Archie's equation from a combination oflogs and core measurements. In Middle East Oil Show 1989 Jan. 1. Societyof Petroleum Engineers; Olsen C, Hongdul T, Lykke Fabricius I.Prediction of Archie's cementation factor from porosity and permeabilitythrough a specific surface. Geophysics. 2008 March; 73(2):E81-7; eachincorporated herein by reference in its entirety).

In a preferred embodiment, the resistivity log is measured in twoperpendicular dimensions: between the points in the vertical directionand between the points across the well. Anisotropy of conductancefollows the anisotropy detected by the NMR signals, and such findingcross-validates both methods.

Resistivity logs are electrical well logs that record the resistivity ofa formation. Resistivity is usually recorded in ohm meters (Ωm) and isdisplayed on, for example, track 4 of a well log. Three depths ofresistivity can be logged (shallow, medium, and deep) that record theresistivity of the formation with increasing distance away from theborehole. Resistivity logs can be interpreted to infer information aboutthe porosity of a formation, the water saturation, and the presence ofhydrocarbons (See Evenick, J. C., 2008, Introduction to well logs &subsurface maps: PennWell Corporation; Asquith, G. and D. Krygowski,2004, Basic well log analysis, second edition: The American Associationof Petroleum Geologists, incorporated herein by reference in entirety).The resistivity logging can be likewise validated at multiple depths anddirections by the inventive NMR method. Non-limiting examples of theresistivity sensors and tools, adapted for well-logging purposes are:Array Laterolog Tool (ALAT), EWR®-PHASE 4 Resistivity Sensor,Micro-Spherical Focused Log (MSFL™) and Microlog (ML) Tools byHulliburton, the tools by Schlumberger, Century Geophysical, WeatherfordInternational, KMS Technologies, CNPC, Daleel, PASI GeophysicsInstruments, OYO Corporation, GeoVista.

The tools intended for exploration of shallow conductivity utilizedirect electric contact and measure resistance to the current at thefixed voltage difference. The tools intended for measuring depthconductivity utilize electromagnetic induction. Induction-conductivitylogging involves the measurement of induced electromagnetic fields inthe rock and resulting eddy currents. Induction tools consist ofmultiple coil arrays designed to optimize vertical resolution and depthof investigation. The basic building block of multiple-coil arrays isthe two-coil sonde. The sensor consists of two coaxial induction coilemitting electromagnetic waves. The processing software subtracts thedirect effects of the emitter coil on the receiver coil. The alternatingfield of the emitter induces eddy currents in the environment which inturn produce the proportional electromagnetic signals detected by thereceiver. The in-depth measurements are especially valuable todifferentiate the invaded zone (permeated by the drilling fluids) andthe virgin zone (where the inherent content of water and hydrocarbon inthe pores more closely resembles the bulk composition in the reservoir).

The results of resistivity logging can be converted in rock permeability(See Wyllie M R J, Rose W D, Some theoretical considerations related tothe quantitative evaluation of the physical characteristics of reservoirrock from electric log data, Trans AIME 189:105p, (1950)—incorporatedherein by reference in entirety). Wyllie and Rose proposed an empiricalequation which is a modification of the Carman-Kozeny equation (SeeCarman P C, Fluid flow through granular beds transactions, Inst Chem EngLond 15:150-166, (1937) incorporated herein by reference) thatsubstituted irreducible water saturation for the specific area term. Asthe specific surface area is quite difficult to measure directly byconventional methods and therefore it is linked with pore size, this inturn controls the irreducible water saturation.

K=P×Φ ^(Q) /S ^(R) _(wir)   (24)

Where:

K—is the permeability of the formation,

Φ—is the porosity,

S_(wir)—is the irreducible water content,

P and R—are the empirical exponent coefficients.

The irreducible water content is a convenient proxy for pore aspectratio, since the water bound to the surface of the pores serves as amarker of the surface (a similar idea to using T₂ relaxation fordetermining pore sizes). This irreducible water content can be foundthrough NMR methods or capillary pressure measurements in laboratoryexperiments (Xiao L, Mao Z Q, Jin Y. Calculation of irreducible watersaturation (S wirr) from NMR logs in tight gas sands. Applied MagneticResonance. 2012 Feb. 1; 42(1):113-25; Working Guide to ReservoirEngineering, 2010, Pages 97-232 in Chapter 2 titled “FormationEvaluation” by William C. Lyons, incorporated herein by reference inentirety). NMR methodology is the most reliable, segregating thefractions by relaxation times. For example, the clay bound water(non-reducible subset, CBW) responds with T₂=0.5-3 msec, capillary boundwater (more mobile irreducible subset, BWI) responds with T₂=3-15 msec,free water responds with T₂=15-100 msec and hydrocarbons respond withT₂>100 msec (See Crain's Petrophysical Handbook, incorporated herein byreference). The fundamental advantage of resistivity logs is the ease ofproducing, and depth attribution of the data, while the disadvantage ismodel-dependence and insufficiency as a stand-alone technique.Resistivity logging enables modeling of porosity, tortuosity,anisotropic variations of the latter and overall permeability,especially when combined with the empirical constants or the inputs ofthe collaborating methods.

Sonic logging is a well logging technique/tool that provides aformation's interval transit time, designated as Δt, which is a measureof a formation's capacity to transmit seismic waves. Geologically, thiscapacity varies with lithology and rock textures, most notablydecreasing with increasing effective porosity. This means that a soniclog can be used to calculate the porosity of a formation if the seismicvelocity of the rock matrix, V_(mat), and pore fluid, and V_(l), areknown.

For a relationship between sound travel time and porosity, the Wyllietime-average equation is the most accepted. According to the equation,the total travel time recorded on the log is the sum of the time thesonic wave spends traveling the solid part of the rock, called the rockmatrix and the time spent traveling through the fluids in the pores.This equation is empirical and makes no allowance for the structure ofthe rock matrix or the connectivity of the pore spaces. The Wyllietime-average equation is:

$\begin{matrix}{\frac{1}{V} = {\frac{\phi}{V_{f}} + \frac{1 - \phi}{V_{mat}}}} & (25)\end{matrix}$

Where:

V—is the measured overall seismic velocity of the formation;

V_(f)—is the seismic velocity of the pore fluid;

V_(mat)—is the seismic velocity of the rock matrix;

ϕ—is porosity.

Pore structure can also be estimated from sonic/acoustic data (see TaoG, King M S. Porosity And Pore Structure From Acoustic Well LoggingData 1. Geophysical Prospecting. 1993 May; 41(4):435-51; Weger R J,Eberli G P, Baechle G T, Massaferro J L, Sun Y F. Quantification of porestructure and its effect on sonic velocity and permeability incarbonates. AAPG bulletin. 2009 Oct. 1; 93(10):1297-317; Sun Y F.Seismic signatures of rock pore structure. Applied Geophysics. 2004 Jul.1; 1(1):42-9; each incorporated herein by reference in its entirety). Ingeneral, smaller pore sizes and more convoluted geometries produce moreboundaries that reflect the soundwave at the same porosity fraction andthe resulting measured velocity is lower for higher complexity systems.Different directions may present variable porosity and boundarycomplexity, accounting for anisotropy.

The comparison between NMR, resistivity and sonic logs indicate thatnone of these techniques is best used as a stand-alone method,especially under the conditions of rapid exploration inlogging-while-drilling conditions. In all individual techniques, theunderlying theoretical assumptions are simplistic, and all methodsrequire empirical adjustments. There is a lack of a comprehensive,precise model able to correctly anticipate the productivity of thereservoir system based on rapid analysis conducted directly in the welland inform the termination of drilling of an exploratory well if theresult is negative. Cutting the exploration risk in each situation ofdecision-making warrants a potentially significant economic effect,considering that most of the hydrocarbon sources are in unconventionalforms that require stimulated production (See Arthur M A, Cole D R.Unconventional hydrocarbon resources: prospects and problems. Elements.2014 Aug. 1; 10(4):257-64, incorporated herein by reference in itsentirety).

The comprehensive permeability modeling of the present disclosureincorporates tortuosity and porosity anisotropies as features of amultivariate regression predictor, linking laboratory-validated valuesof permeability and laboratory-validated structural parameters ofdiverse formations with the NMR, resistivity and sonic parameterscollected in well-logging by an integrated sensor unit. The mutuallycomplementing character of the individual methods warrants that thecoefficients of the regression model use this synergy and provide asuperior analysis at a comparable cost of real-time data collection.

Preferably, the first step in this methodology is the collection of rockcores such as sandstone or carbonate cores. Different types of porousrocks are analyzed separately. Commercial providers manufacture and sellthe cylindrical rock cores representing hydrocarbon reservoirs: KocurekIndustries INC., Vinci Technologies, Rockman, Bureau Veritas CommoditiesCanada Ltd.

Core Research Center was established to coordinate these efforts andpreserves valuable rock cores for scientists and educators fromgovernment, industry, and academia. Other core depositories includeAlabama Geological Survey State Oil and Gas Board Core Warehouse,Alabama Geological Survey State Oil and Gas Board Core Warehouse, AlaskaGeologic Materials Center, Alaska Geologic Materials Center OnlineInventory, Arizona Geological Survey (AZGS) 1993 Core Repository Report,Arkansas Geological Survey Norman F. Williams Well SampleLibrary,California Well Sample Repository, Connecticut Geological SurveyBedrock Core Repository, Delaware Geological Survey Outer ContinentalShelf Core and Sample Repository, Florida Geological Survey Core andCuttings Repository, Illinois State Geological Survey Geological SamplesLibrary, Search Illinois Geological Samples Library, Iowa GeologicalSurvey Oakdale Rock Library and Research Facility, Kansas GeologicalSurvey Kansas Core Library, Kansas Geological Survey KansasRotary-cutting samples, Kentucky Geological Survey Well Sample and CoreLibrary, Well Sample and Core Library Database Search, LACCORE NationalLacustrine Core Repository, Louisiana Geological Survey Resource CenterCore Repository, Maine Geological Survey Core Repository and ExplorationRecords, Michigan Geological Repository for Research and Education,Minnesota Department of Natural Resources Division of Lands and MineralsDrill Core Library, Mississippi Department of Environmental Quality,Environmental Geology Division, Office of Geology, Core and SampleLibrary Missouri Department of Natural Resources McCracken Core Libraryand Research Center Nebraska Conservation and Survey Division GeologicalSample Repository, Nevada Bureau of Mines and Geology Great BasinScience Sample and Records Library, New Mexico Subsurface Data and CoreLibraries, North Carolina Geological Survey Coastal Plain Office CoreRepository, North Dakota Geological Survey Wilson M. Laird Core andSample Library, Ohio Department of Natural Resources Geological SurveyCore and Sample Repository, Oklahoma Geological Survey Core and WellCutting Research Facility, Pennsylvania Department of Conservation andNatural Resources (DCNR), South Carolina Geological Survey CoreRepository, South Dakota Geological Survey Core and Cuttings Repository,Core and Cuttings Repository Database, Texas Bureau of Economic GeologyCore Research Facilities, Integrated Core and Log Database, UtahGeological Survey Core Research Center, Wisconsin Geological & NaturalHistory Survey Research Collections and Education Center (CoreRepository).

The USGS maintains the most diverse public-access core collections inthe USA. A variety of core sub-collections are available in therepository, including those from oil shale development; Eniwetok Atoll;Cajon Pass, Calif.; Yellowstone Park; and off-shore wells. In addition,CRC curates collections of cuttings (rock chips) brought to the surfaceduring drilling operations. The core and cuttings collection is alsoaccompanied by a large collection of thin sections, which are used toexamine microscopic details of the rocks. Images of the thin sectionsand photographs of some cores are available for viewing and download.Files containing chemical and physical analyses, core descriptions,stratigraphic charts, and various other analyses performed by previoususers of the collection can also be downloaded. The CRC houses about 2million feet of core in the general collection of petroleum explorationand development holes as well as in specialized collections. These corescome from 33 states and about 95 percent were donated by petroleum andmining companies, State geological surveys, other Federal agencies, anduniversities; about 5 percent are special scientific cores drilled bythe USGSAnalogous infrastructure exists outside of the USA, withoutlimitation: Kochi Core Center Kochi University, Japan and the Universityof Bremen, Germany. The samples are available via an application processand released by a decision of a research board.

The cylindrical cores are sealed in a high-pressure testing cell and aresubjected to permeability test using a fluid with known viscosity orwith a viscosity profile emulating natural hydrocarbon (another sampleof heavy oil, cracking residue, tar). Alternatively, gas or floodingbrine permeability is measured. Prior to pressing through the core, thefluid (or gas) is equilibrated with water (or water vapor) and passes apre-filter eliminating potential sediments (dust). The core is wetted bywater to the extent matching the non-reducible level or to a fixed levelof water content. The permeability test is repeated at 2 watersaturation levels and with several oil (gas) models. After completion ofthe realistic permeability test, the wetted cores with the modelhydrocarbons entrapped in the pores together with water are subjected toNMR, resistivity and sonic characterization. The NMR examinationincludes measuring overall signal strength at different chemical shifts,measuring of relaxation times for all components at differentorientations of the sample, measuring water restricted diffusioncoefficients, measuring diffusion anisotropy, computing porosity, poresize distribution and tortuosity based on NMR data. The core nextundergoes resistivity studies at the initial and varied water contents,with the several directions of the current, to measure the anisotropy offormation factor, resistivity index and the empirical coefficients n andm in the Archie's Law form. The core also undergoes acoustic studies atseveral water contents and directions. After completing these tests, thecore is repeatedly extracted by hot diesel oil (to prevent asphaltenedeposition), the diesel oil is displaced and dissolved by heptane, andthe core is dried. The dried core is subjected by a detailed porometricstudy and “true” tortuosity determination by xenon NMR or microscopy(See Wang R, Pavlin T, Rosen M S, Mair R W, Cory D G, Walsworth R L.Xenon NMR measurements of permeability and tortuosity in reservoirrocks. Magnetic resonance imaging. 2005 Feb. 1; 23(2):329-31; Albers B,Wilmanski K. Acoustics of two-component porous materials withanisotropic tortuosity. Continuum Mechanics and Thermodynamics. 2012Nov. 1; 24(4-6):403-16; Wang R, Mair R W, Rosen M S, Cory D G, WalsworthR L. Simultaneous measurement of rock permeability and effectiveporosity using laser-polarized noble gas NMR. Physical Review E. 2004Aug. 31; 70(2):026312; incorporated herein by reference in entirety).

The procedure is repeated for multiple samples, e.g., separately forcarbonates and sandstones. The number of samples included in thetraining set for each group is not less than 50, preferably 75, evenmore preferably >100. For each sample, multiple compositions areanalyzed (water contents and oil models), thus the training set mayinclude 1000-2000 data rows, relating the high-throughput loggingparameters to the observed permeabilities (wettability, mechanicalstrengths). For cost-efficiency, the training sets are expected to bemaximally diversified to ensure that they represent the entire varietyof hydrocarbon cores. In the most preferred embodiment, the coresoriginating in different parts of the globe are procured forcharacterization.

At the next step, the parameters measured by high-throughput loggingtools such as NMR, resistivity and acoustic probes are fit to the truevalues of permeability, porosity and tortuosity obtained in laboratoryexperiments. Multiple fitting methods are possible, with multipleconvergence and residual minimization criteria (the Least Square Methodis one non-limiting embodiment).

In one non-limiting embodiment, the following expression is fitted tothe experimental permeability:

A ₁×[τ_(NMR) ^(B1)]×[τ_(RES) ^(B2)]×[Φ_(NMR) ^(B3)]×[Φ_(RES)^(B4)]×[Φ_(SONIC)^(B5)]×[(τ_(NMRZ))^(B6)/(τ_(NMRXY))^(B6)]×[(τ_(RESZ))^(B7)/(τ_(RESXY))^(B7)]×[(Φ_(SONICZ))^(B8)/(Φ_(SONICXY))^(B8)]×S_(w) ^(B9) ×S _(wir) ^(B10) ×R ^(B11) =K   (26)

Where:

-   -   K—is permeability, wherein v=(K/μ)×ΔP/ΔL (the volume flow        velocity v [m/sec] is directly proportional to the pressure        gradient ΔP/ΔL [kg/sec²×m²] and structural permeability K [m² or        Darcy] and is inversely proportional to dynamic viscosity μ        [kg×m×s⁻¹]    -   A₁—is the empirical proportionality factor,    -   [τ_(NMR) ^(B1)]—is the diffusional tortuosity determined by NMR        and B1 is the respective exponent of the fitting model to be        determined in the training process.    -   [τ_(RES) ^(B2)]—is the tortuosity determined by resistivity (or        Formation Factor F) and B2 is the respective exponent of the        fitting model to be determined in the training process.    -   [Φ_(NMR) ^(B3)]—is the overall porosity determined by NMR and        the respective fitting coefficient B3.    -   [Φ_(RES) ^(B4)]—is the overall porosity determined by        resistivity logging and the respective fitting coefficient B4.    -   [Φ_(SONIC) ^(B5)]—is the overall porosity determined by acoustic        probing and the respective fitting coefficient B5.    -   [(τ_(NMRZ))^(B6)/(τ_(NMRXY))^(B6)]—is the ratio of diffusional        tortuosity in the direction Z to one measured in the plane XY        and the respective fitting coefficient B6.    -   [(τ_(RESZ))^(B7)/(τ_(RESXY))^(B7)]—is the ratio of resistivity        tortuosity (formation factors) measured in the direction Z to        one measured in the plane XY and the respective fitting        coefficient B7    -   [(Φ_(SONICZ))^(B8)/(Φ_(SONICXY))^(B8)]—is the ratio of porosity        determined by acoustic probing in the direction Z to the        porosity determined by the acoustic probing in the plane XY,        wherein B8 is the respective fitting coefficients.    -   S_(w) ^(B9)—is the water saturation fraction with the respective        fitting factor.    -   S_(wir) ^(B10)—is the irreducible water content with the        respective fitting factor.    -   R^(B11)—resistivity at the given water saturation.

All definitions above apply to water flooding the innermost 0-5 mm ofthe borehole wall, the closest to the drilling in progress.

The equation (26) outwardly resembles the approach of Wyllie and Rose(equation (24) or Archie's Law. However, it principally differs byincorporation of the the anisotropies determined via diffusionaltortuosity, measuring formation factors in different directions andacoustic probing in different directions. The tortuosity anisotropiesreflected in (26) extend to porosities as well. All aspects of all 3logging methods synergize in the formula (26), allowing a flexiblenon-linear fit to the training sample.

The form (26) is maintained for all N experimental series comprising thetraining set. The fitting coefficients A1 and B1-B11 are extracted froma Least Square relationship minimizing the residual between thepredicted permeability based on these rapid logging metrics and the realmeasured permeability. An expression analogous to (26) can be writtenfor other parameters of interest characterizing a rock. One of suchparameters is water content, including irreducible water content.Another parameter is true porosity, true tortuosity, true fraction ofcul-de-sacs, true anisotropy. Still another parameter, withoutlimitation is wettability and yet another is fracturing gradient.Different sets of fitted regression coefficients link thehigh-throughput logging data and any true laboratory-measured dataimportant for hydrocarbon production.

In a preferred embodiment, the 1000-2000 data points for all regimes inthe training set are subdivided in the true training (50% of data) andthe testing set (50% of data). The testing set is not included in theregression models extracting the coefficients A and B fitting the modelsto reality. Instead, the measurements conducted for the testing set areadjusted by the fitting coefficients extracted using the true trainingset. If the prediction is generalizable, the high throughputmeasurements in the testing set organized according to the rulesestablished in the true training set predict the observed permeabilitieswith the same quality as in the true training set. Lack of agreementbetween the true training and the testing sets questions thecomputational approach and a different mathematical form is required toachieve reproducibility for all locations and geological histories. Themost valuable models are those that correctly predict the parameters inthe samples very different from the training set. As an example, anideal training process conducted in calcite cores extracted in the USAshould also correctly predict the results in dolomite or aragonite coresextracted in China without limiting. In this sense, the form (26) isonly preliminary and other forms (“kernels”) and convergence criteriaare conceivable to achieve wide reproducibility of predictions (SeeAmari S I, Wu S. Improving support vector machine classifiers bymodifying kernel functions. Neural Networks. 1999 Jul. 1; 12(6):783-9,incorporated herein by reference in entirety).

In a preferred embodiment, the form (26) is fitted to fracturingpressure gradient. The mechanical properties of the rock within the samemineralogical category correlate with permeability, porosity, poredistribution and geometry, acoustic properties, water absorption (SeeKoceir M, Tiab D. Influence of stress and lithology on hydraulicfracturing in Hassi Messaoud reservoir, Algeria. InSPE/AAPG WesternRegional Meeting 2000 Jan. 1. Society of Petroleum Engineers; ShahbaziK, Abdideh M, Hadipoor M. Modelling hydraulic fracturing process in oneof the Iranian southwest oil reservoirs. Applied Earth Science. 2017Jul. 3; 126(3):108-17; Santamaria J C, Velez E I, Bahamon J I.Differential Cased-Hole Shear Anisotropy (DCHSA) Methodology as aCalibration and Evaluation Tool of Hydraulic Fracture Height inColombian Fields. InSPE Heavy and Extra Heavy Oil Conference: LatinAmerica 2014 Sep. 24. Society of Petroleum Engineers, incorporatedherein by reference in entirety). The empirical correlation (26)calculates tensile stresses in the rock formation, by fitting thecomponent data to the hydraulic pressures causing fragmentation of thecores in the laboratory conditions, with the adjustments to the scalefactor. The resulting set of fitting coefficients A1, B1-B11 is verydifferent for each predicted parameter (fracturing pressure,wettability, permeability).

In a preferred embodiment, the results of the training studies areorganized in an instruction set for a processor controlling themulti-sensor downhole assembly. The software optionally sends therecommendations to the human operators in response to the estimatedpermeability, wettability and hydrocarbon content that is compared withthe economically viable cutoff. Based on the instant feedback, theexpensive drilling process may be terminated if the projected depthexpected for the hydrocarbon reservoir is reached. In case ofsubstantial productivity, the acquired data can also be used to computethe position of maximal yield for branching and identify an extent offracturing (and the stabilizing hydraulic pressure) that is the mosteconomically viable.

In a further preferred non-limiting embodiment, the process and systemof the present disclosure rely on fitting models applied to themeasurements produced during drilling or after drilling with variableintervals of time after drilling. In another non-limiting variation, thefitting model (26) can be simplified to rely only on the restricteddiffusion parameters and to omit the resistivity and sonic contributions(see the Examples). In a still further non-limiting variation, the model(26) is made more complex by including time-dependent terms fordiffusion tortuosity measured by the NMR sequences of variable length.In a further non-limiting variation, the model is made more complex byincluding detergent, salinity, viscosity, temperature, pressure andturbulence effects, emulating the broad range of parameters in aborehole. The number of data rows in the training set as envisioned inthis invention by far exceeds the number of potential variables,excluding the possibility of an over-fit. However, working withsimplified embodiments of the general fitting model (26) presents theadvantages of easier classification and delineation of the functionalregimes.

In further preferred non-limiting embodiments, the process and system ofthe present disclosure can be used to predict, design, calculate and/orfacilitate any of:

pore network complexity and permeability anisotropy based on NMRdiffusion measurements;

permeability anisotropy in the laboratory as well as in the field usingNMR equipment or NMR logging tools;

the perforation scheme in cased hole vertical wells towards the maximumpermeability value in the horizontal direction;

the perforation scheme in cased hole directional wells towards themaximum permeability value in the horizontal direction;

the perforation scheme in case hole horizontal wells towards the maximumpermeability value in the horizontal direction;

the permeability profile in anisotropic porous media in carbonatereservoirs in vertical wells;

the permeability profile in anisotropic porous media in carbonatereservoirs in horizontal wells;

the permeability profile in anisotropic porous media in sandstonereservoirs in vertical wells;

the permeability profile in anisotropic porous media in sandstonereservoirs in horizontal wells;

the completion scheme in horizontal wells, extended reach wells, maximumcontact wells for perforated casing completion type by identifying themaximum permeability direction in which the casing can be perforated;

the completion scheme in horizontal wells, extended reach wells, maximumcontact wells for slotted liner casing completion type by identifyingthe maximum permeability direction in which the slots can be located tomaximize the well production;

the stimulation treatment in heterogeneous long extended reach carbonatereservoirs by identifying the high permeability section and lowpermeability sections that exactly need the acid stimulation process;

the filter cake treatment in heterogeneous long extended reach carbonatereservoirs by identifying the high permeability section and lowpermeability sections that exactly need the acid stimulation process;

the filter cake treatment in heterogeneous long extended reach sandstonereservoirs by identifying the high permeability section and lowpermeability sections that exactly need the acid stimulation process;

the multi-stage fracturing scheme in tight sandstone to locate thefracture across the less isotropic high permeability sections of thehorizontal well;

the multi-stage fracturing scheme in tight carbonate to locate thefracture across the less isotropic high permeability sections of thehorizontal well; and

the multi-stage fracturing scheme in unconventional shale to locate thefracture across the less isotropic high permeability sections of thehorizontal well.

Having generally described this disclosure, a further understanding canbe obtained by reference to certain specific examples which are providedherein for purposes of illustration only and are not intended to belimiting unless otherwise specified.

EXAMPLE 1 Sample Preparation and Conducting of NMR

Rock samples: Four Limestone rock samples were cored to have acylindrical geometry of 2 inch length and 1.5 inch diameter. Both coreswere fully saturated with de-ionized water plus 0.2 g/L of HCl. Thepetrophysical parameters of the rock core plugs are summarized inTable 1. The samples have variable porosity and permeability values. Theobserved permeability does not correlate with porosity as such. Forexample, Pink Desert sample has lower permeability than Indiana (2) orD1 despite higher porosity. Since porosity is the most used predictor ofpermeability according to Kozeny-Carman approach, this lack ofcorrelation with porosity is significant and calls for more diversecorrelates between the log data and permeability.

TABLE 1 Petrophysical Properties of the Samples Sample Name Porosity (%)Permeability (mD) Indiana (1) 10.81 0.73 Indiana (2) 17.43 216.89 PinkDesert 24.99 31.96 D1 17.06 350

Pulsed Field Gradient NMR (PFG NMR): An NMR experiment with a timedpulse with spatial-dependent field frequency is called a pulsed fieldgradient NMR. There are four characteristics by which the gradient isidentified:

(1) Axis in which the gradient is applied

(2) Strength of the gradient

(3) The shape of the applied gradient

(4) Duration of the gradient application.

PFG NMR Techniques are the basics of magnetic resonance imaging (MRI),and of diffusion studies by nuclear magnetic resonance. Diffusion isusually similar in every direction for most liquids and homogenous solidmaterials such as gels. Such materials are referred to as isotropic andhave a single diffusion coefficient (D). On the other hand, reservoirrock and biological tissues are highly heterogeneous and typically havedifferent coefficients of diffusion in different directions. PFG NMRmethods are especially useful in the study of diffusion anisotropy, asmentioned earlier; here this method was applied to rocks to investigatethe correlation between diffusion tortuosity and permeability. PFG-NMRexperiments were carried out on an Oxford Instruments NMR Rock CoreAnalyzer MARAN DRX 12.2 MHz coupled with a 5.3-cm I.D radiofrequencycoil and three dimensional (Gx, Gy, and Gz) magnetic field gradientswith strength up to 30 G/cm. The 13-interval Bipolar Gradients pulsesequence (APFGSTE) was applied to avoid unnecessary signal attenuation.This can prevent significant error in the measurements of the restricteddiffusion coefficient, D_(R), due to the susceptibility difference,which causes internal magnetic field gradients. The signal attenuation(S/S₀) for the 13-interval APGSTE sequence is given by:

$\begin{matrix}{\frac{S}{S_{o}} = {\exp\left\{ {- {\quad{D_{R}\gamma^{2}\left. \quad\left\lbrack {{{\delta^{2}\left( {{4\Delta} + {6\lambda} - \frac{2\delta}{3}} \right)}g_{a}^{2}} + {2{{\lambda\delta}\left( {\delta_{1} - \delta_{2}} \right)}g_{a}g_{0}} + {\frac{4}{3}\lambda^{3}g_{0}^{2}}} \right\rbrack \right\}}}} \right.}} & (27)\end{matrix}$

Where:

-   -   S₀ is the NMR signal in the absence of applied magnetic field        gradients (μV).    -   D_(R) is the Restricted self-diffusion coefficient of the fluid        (m²/s),    -   Δ is the duration between two applied magnetic field ‘Diffusion        Time’ (msec),    -   δ is the duration of the applied magnetic field gradient (msec).    -   γ is the gyromagnetic ratio of the nucleus being studied        (=2.68×10⁸ Hz/T for ¹H nucleus).    -   g_(a) is the strength of the applied magnetic field gradient        (T/m),    -   g_(o) is strength of the internal magnetic field gradient (T/m).    -   λ is the time between the first two (π/2) RF pulses (msec), and    -   δ₁ and δ₂=pre- and post-pulse time respectively (msec).

By choosing the same value for the two time periods δ₁ and δ₂ andselecting diffusion time larger than the time between the first two(π/2) RF pulses, the previous equation can be expressed as follows(equation 28), enabling D_(R) to be measured from the slope of thenatural logarithm of the signal attenuation versus −b′.

$\begin{matrix}{\frac{S}{S_{o}} = {\exp\;\left\{ {- {\quad{{D_{R}\gamma^{2}\left. \quad\left\lbrack {{\delta^{2}\left( {{4\Delta} + {6\lambda} - \frac{2\delta}{3}} \right)}g_{a}^{2}} \right\rbrack \right\}} = {{{\exp\left( {{- {D_{R}\left( {2{\delta\gamma}\; g_{a}} \right)}^{2}}\left( {\Delta - {\frac{3}{2}\lambda} - \frac{\delta}{6}} \right)} \right)}\mspace{20mu}{where}\mspace{14mu} b^{\prime}} = {\left( {28{\gamma g}_{a}} \right)^{2}\left( {\Delta - {\frac{3}{2}\lambda} - \frac{\delta}{6}} \right)}}}}} \right.}} & (28)\end{matrix}$

Error! Reference source not found.3 shows a representation of how theNMR gradient can be utilized to measure the anisotropic characteristicof the rock samples. If the z-direction gradient is applied, themovement of the molecules will be directed in the vertical direction ofthe sample, which gives an indication of the vertical permeability ofthe rock sample. On the other hand, if the x-direction gradient isapplied, the movement of the molecules will be directed in thehorizontal (radial) direction. These measurements are performedseparately in order to have a single value of diffusion in eachdirection. The diffusion tortuosity in any experiment is calculated thenby using the data at diffusion time that is large enough to reach thelong-term asymptotic regime, as shown in FIG. 14. The relativedifference in diffusion coefficients or τ_(d) measured in differentdirections is hypothesized to give the degree of pore geometryanisotropy for the porous media. In order to use this concept forevaluation of permeability anisotropy, diffusion tortuosity is proposedto correlate permeability.

EXAMPLE 2 Identification of the Sufficient Time-Frame for Measurement ofRestricted Diffusion Coefficients

FIGS. 14 and 15 shows the inverse of diffusion tortuosity (D_(R)/D₀) asa function of observation time for permeable and tight limestone,respectively. The results illustrate the need of planning an NMRsequence such that the 1H nuclei have time to diffuse to the pore walls.The time interval is shorter for smaller pores. The correct choice oftiming is important since the ratio of the restricted and the bulkdiffusion coefficients is a strong predictor of permeability and a2-fold error in this metric can lead to orders of magnitude error inpermeability (FIG. 16).

EXAMPLE 3 Diffusional Tortuosity Strongly Correlates with PermeabilityAcross Different Cores with Diverse Structure

FIG. 16 shows the relationship between the permeability of the samplesand the diffusion tortuosity. Permeability increases as diffusiontortuosity decreases. This can be explained by the pore size and networkcomplexity; sample A has higher diffusion tortuosity since it hassmaller pore sizes and more complex pore network compared to sample B(FIG. 17). The strong relationship between permeability and diffusiontortuosity suggests that permeability anisotropy correlates to diffusiontortuosity anisotropy. The latter can be detected using the PFG NMRapproach in the laboratory or potentially in the field (NMR loggingtool). That is, in a rock made of alternating tight and permeablelayers, the vertical permeability (and thus τ_(d)) are different fromthe horizontal value. While the support by resistivity and sonic loggingis important, the results of FIG. 16 indicate that diffusionaltortuosity alone can be a strong individual predictor of permeability inat least some combinations of rock and hydrocarbon compositions. In thisindividual mode, the prediction is performed by relating each tortuosityindex τ_(z) and τ_(xy) to directional permeability by the formula:

K _(z) =C ₁×(τ_(z))^(C2) +C ₃ ; K _(xy) =C ₄×(τ_(xy))^(C5) +C ₆   (29)

wherein

-   C₁-C₆ are the empirical coefficients;-   K_(z), K_(xy) are the directional permeability values [m²],-   τ_(z), τ_(xy) are the directional diffusional tortuosity values.    To summarize, the advantage of using NMR compared to imaging or    extraction of different cores is two-fold: a) it is non-invasive and    requires less time; b) it can be applied in the field.

1. A downhole method of determining permeability anisotropy with NMR ina borehole, comprising: first, inserting an NMR apparatus into theborehole which is located in a hydrocarbon-containing geologicalformation, then a) determining a first ¹H self-diffusion coefficient(D_(b)) of bulk water in a portion of the hydrocarbon-containinggeological formation where the water is present outside of the pores, byexposing the out of the pore portion of the hydrocarbon-containinggeological formation to an NMR pulse gradient along an axis Z of theportion of the hydrocarbon-containing geological formation; wherein theout-the pore portion comprises brine flooding the borehole, wherein thebrine is measured by NMR prior to flooding, post-flooding samples of thebrine extracted from the borehole; b) determining a second ¹Hself-diffusion coefficient (D_(R)) of water in a portion of thehydrocarbon-containing geological formation in a restricted porousenvironment by an NMR sequence with the magnetic pulse gradient alongthe axis Z; selecting the timescale of performing the NMR sequence in(a) and (b) such that the timescale corresponds to the asymptoticminimal values of the restricted diffusion coefficients for the entirepore size distribution; d) calculating a diffusion tortuosity valueτ_(z) as τ_(z)=D_(b)/D_(R); e) repeating steps (a)-(d) with the magneticpulse gradient along any directions in a plane XY perpendicularly to theaxis Z of the hydrocarbon-containing geological formation and computinga diffusion tortuosity metric τ_(xy); f) determining the permeability ofthe hydrocarbon-containing geological formation based on a ratio of aplurality of diffusional tortuosity values measured in differentdirections.
 2. The method of claim 1, wherein the NMR-measureddiffusional tortuosity is the only parameter used to predictpermeability by relating each tortuosity index τ_(z) and τ_(xy) todirectional permeability by:K _(z) =C ₁×(τ_(z))^(C2) +C ₃ ; K _(xy) =C ₄×(τ_(xy))^(C5) +C ₆; whereinC₁-C₆ are empirical coefficients; K_(z), K_(xy) are directionalpermeability values [m²], τ_(z), τ_(xy) are directional diffusionaltortuosity values.
 3. The method of claim 1, wherein the NMR pulsegradient is an APGASTE 13 sequence of formula:$\frac{S}{S_{o}} = {\exp\left\{ {- {\quad{D_{R}\gamma^{2}\left. \quad\left\lbrack {{{\delta^{2}\left( {{4\Delta} + {6\lambda} - \frac{2\delta}{3}} \right)}g_{a}^{2}} + {2{{\lambda\delta}\left( {\delta_{1} - \delta_{2}} \right)}g_{a}g_{0}} + {\frac{4}{3}\lambda^{3}g_{0}^{2}}} \right\rbrack \right\}}}} \right.}$wherein S₀ is the NMR signal in the absence of applied magnetic fieldgradients (μV). D_(R) is the restricted self-diffusion coefficient ofthe fluid (m²/s), Δ is the duration between two applied magnetic field‘Diffusion Time’ (msec), δ is the duration of the applied magnetic fieldgradient (msec). γ is the gyromagnetic ratio of the nucleus beingstudied (=2.68×10⁸ Hz/T for ¹H nucleus), g_(a) is the strength of theapplied magnetic field gradient (T/m), g_(o) is the strength of theinternal magnetic field gradient (T/m), λ is the time between the firsttwo (π/2) RF pulses (msec), and δ₁ and δ₂=pre- and post-pulse timerespectively (msec).
 4. The method of claim 3, further comprising: theself-diffusion coefficient of water from hydrocarbon contributions inthe NMR measurement with DOSY.
 5. The method of claim 1, wherein one ormore of the determining steps is carried out with a downhole sensorassembly comprising an NMR apparatus.
 6. The method of claim 5, whereinthe downhole sensor assembly further measures resistivity and acousticlogs. 7-11. (canceled)